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Hiotographic 

Sciences 

Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  145*0 

(716)  872-4503 


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CIHM/ICMH 

Microfiche 

Series. 


CIHIVI/ICIVIH 
Collection  de 
microfiches. 


Canadian  Institute  for  Historical  Microreproductions  /  Institut  canavdien  de  microreproductions  historiques 


Technical  and  Bibliographic  N\  tas/Notes  techniques  at  bibliographlques 


The  Iratitute  has  attempted  to  obtain  the  best 
original  copy  available  for  filming.  Features  of  this 
copy  which  may  be  bibliographically  unique, 
which  may  alter  any  of  the  images  in  the 
reproduction,  or  which  may  significantly  change 
the  usual  method  of  filming,  are  checked  below. 


D 


D 


D 
D 


D 


D 


Coloured  covers/ 
Couverture  de  couleur 


I     I    Covers  damaged/ 


Couverture  andommag6e 

Covers  restored  and/or  laminated/ 
Couverture  restaurte  et/ou  pellicula 


I      I    Cover  title  missing/ 


Le  titre  de  couverture  manque 

Coloured  maps/ 

Cartes  giographiques  en  couleur 

Coloured  ink  (i.e.  other  than  blue  or  black)/ 
Encre  de  couleur  (i.e.  autre  que  bleue  ou  noire) 

Coloured  plates  and/or  illustrations/ 
Planches  et/ou  illustrations  en  cooleur 


Bound  with  other  material/ 
RelM  avec  d'autres  documents 

Tight  binding  may  cause  shadows  or  distortion 
along  interior  margin/ 

Lareliure  serrAe  peut  causer  de  i'ombre  ou  de  la 
distortion  le  long  de  la  marge  intArieure 

Blank  leaves  added  during  restoration  may 
appear  within  the  text.  Whenever  possible,  these 
have  been  omitted  from  filming/ 
II  se  r*ut  que  certaines  pages  blanches  ajouttes 
lors  d'une  restaura'Jon  apparaissent  dans  le  texte, 
mais,  ior^que  cela  Atait  possible,  cas  pages  n'ont 
pas  its  filmtes. 

Addi'tional  comments:/ 
Commentaires  supplAmentaires; 


T 
t( 


L'Institut  a  microfilm^  le  meilleur  exemplaire 
qu'ii  lui  a  iti  possible  de  se  procurer.  Les  details 
de  cet  exemplaire  qui  sont  peut-Atre  uniques  du 
point  de  vue  bibliographique,  qui  peuvent  modifier 
une  image  reproduite,  ou  qui  peuvent  exiger  une 
modification  dans  la  methods  normale  de  filmage 
sont  indiqute  ci-dessous. 


n 


/ 


D 
D 


Coloured  pages/ 
Pages  de  couleur 

Pages  damaged/ 
Pages  endommagtes 


□   Pages  restored  and/or  laminated/ 
Pages  restaurtes  et/ou  pelliculAes 

0   Pages  discoloured,  stained  or  foxed/ 
Pages  dAcoiortes,  tachettes  ou  piqu6es 

□    Pages  detached/ 
Pages  d6tach6es 


Showthrough/ 
Transparence 


I     I    Quality  of  print  varies/ 


Quality  InAgaie  de  I'impression 

Includes  supplementary  material/ 
Comprend  du  material  suppl^mentaire 


Only  edition  available/ 
Seule  Edition  disponible 

Pag^  wholly  or  partially  obscured  by  errata 
slips,  ti<sues,  etc.,  have  been  refilmed  to 
ensure  the  best  possible  image/ 
Les  pages  totalement  ou  partiellement 
obscurcies  par  un  feuliiet  d'errata,  une  pelure, 
etc.,  ont  At*  filmAes  d  nouveau  de  fapon  A 
obtenir  la  meilleure  image  possible. 


T 

P 
o 
fi 


O 
b 
t> 
si 
o 
fi 
si 
oi 


Tl 
si 
Tl 
w 

M 
di 
er 
b< 

rlj 
re 
m 


This  item  is  filmed  at  the  reduction  ratio  checked  below/ 

Ce  document  est  film*  nu  caux  de  rMuction  indiquA  ci-dessous. 


10X 

14X 

18X 

22X 

26X 

30X 

V 

12X 


16X 


20X 


24X 


28X 


a2X 


The  copy  fllm«d  h«r«  has  bMn  r«produc«d  thank* 
to  tho  gonorotity  of: 

Douglas  Library 
Quaan's  Univoraity 

Tha  imagaa  appaaring  hara  ara  tha  baat  quality 
poaaibia  conaidaring  tha  condition  and  lagibility 
off  tha  original  copy  and  in  kaaping  with  tha 
ffilming  contract  ipacifications.  . 


Original  copies  in  printad  papar  covara  ara  filmad 
beginning  with  tha  front  covar  and  anding  on 
tha  laat  paga  with  a  printad  or  illuatratad  impras- 
sion,  or  tha  back  covar  whan  appropriata.  All 
othar  original  copiati  ara  filmad  beginning  on  tha 
first  paga  with  a  printad  or  illuatratad  impras- 
sion,  and  aiding  on  tha  last  paga  with  a  printad 
or  illuatratad  imprassion. 


Tha  last  racordad  frama  on  aach  microficha 
shall  contain  tha  symbol  — ^>  (moaning  "CON- 
TINUED"), or  tha  symbol  V  (moaning  "END"), 
whichavar  applies. 

Maps,  plates,  charts,  etc.,  may  be  filmed  at 
different  reduction  ratios.  Those  too  large  to  be 
entirely  included  in  one  exposure  are  filmed 
beginning  in  the  upper  left  hand  corner,  left  to 
right  and  top  to  bottom,  as  many  frames  as 
required.  The  following  diagrams  illustrate  the 
method: 


L'exemplaira  film*  f ut  reprodult  grice  A  la 
g4n4rosit4  de: 

Douglas  Library 
Queen's  University 

Lee  imagaa  suivantes  ont  4tA  raproduites  avac  la 
plus  grand  soin,  compta  tenu  de  la  condition  et 
de  la  nattet*  da  I'exemplaira  film*,  et  en 
conformiti  avac  las  conditions  du  contrat  de 
filmage. 

Lea  axemplaires  originaux  dont  la  couverture  en 
papier  eat  ilmprimAa  aont  ffilmte  an  commenpant 
par  la  premier  plat  et  en  terminant  soit  ppr  la 
darnlAre  page  qui  comporte  une  empreinte 
d'impreaaion  ou  d'illustration,  soit  par  la  second 
plat,  aaion  le  cas.  Tous  las  autres  exemplairas 
originaux  sont  ffilmte  9n  commenpant  par  la 
pramlAre  paga  qui  comporte  une  empreinte 
d'impreaaion  ou  d'illustration  et  en  terminant  par 
la  darniire  page  qui  comporte  une  telle 
empreinte. 

Un  des  symboles  suivants  appars!tra  sur  la 
dernidre  image  de  cheque  microfiche,  selon  le 
caa:  le  symbols  -^  signifie  "A  SUIVRE",  le 
symbols  V  signifie  "FIN". 

Lea  certes,  planches,  tableaux,  etc.,  peuvent  Atre 
fiimAs  A  des  taux  de  rMuction  difff^rents. 
Lorsque  le  document  eit  trop  grand  pour  Atre 
reprodult  en  un  seul  clich*,  il  est  ffiim*  A  partir 
de  Tangle  supArieur  geuche,  de  gauche  A  droite, 
et  de  haut  en  baa,  en  prenant  In  nombre 
d'imagas  n6cessaire.  Lea  diagrammes  suivants 
illustrent  la  mAthoda. 


1 

2 

3 

1 

2 

3 

4 

5 

6 

s       t 


>   '» 


\ 


V 


»     •       %      %    V     \ 


^     V 


SCIENTIFIC  MEMOIRS 


BDJTED  BT 


J.  S.  AMES,  Ph.D. 

PHOMBBBOR    OF    PHYSICS    IN    JOHNS    HOPKINS    DNIVBRSITT 


IX. 


THE  LAWis  OF  GRAVITATION 


THE 


LAWS   OF   GRAVITATION 


MEMOIRS     BY     NEWTON,    BOUGUER 
AND    CAVENDISH 

TOOETUER  WITH  ABSTRACTS  OF  OTHER 
IMPORTANT  MEMOIRS 


TRANSLATED   AND   EDITED   BY 

A.  STANLEY  MACKENZIE,  Pn.D. 

PROKKSSOR  OF  PHYSICS  IN  BRYN  MAWR  COLLKOJC 


NEW    YORK   •:•   CINCINNATI   •:•   CHICAGO 

AMERICAN    BOOK    COMPANY 


V  C-  \-i  s-  M  \ 


>/ 


1/ 


I  I 


Copyright.  1900.  by  Ahkrican  Book  Cokpant. 

W.   p.  I 


^  ' 


0-    '\-4-\\ 


■jr^iSfa^r*.  .. 


196101 


^^KNKRAL    CO.VTKNT.S 


I  I 


n  '.Smplno,..  ekeiel,  of  B„„g,  e       3" 

TI»Bc.rtierc„,„r„vo«j..     *      ,^ 

Hislonciil  account  of  h.«  .     

'^CX''"''''''^-'''--''---:-::-::- '" 

Index...         '43 

H/i 

157 


PREFACE 


P'fi . 


en- 


PA(iK 
V 

I 

0 
19 

2;} 

44 
47 
58 
50 
107 

111 
143 
145 
157 


In  propariiij;  this  volume,  the  ninth  in  the  Scientific  Me- 
moirs series,  the  editor  has  had  in  mind  the  fact  that  the  most 
important  of  the  memoirs  liere  (hnilt  with,  that  of  Cavendish, 
is  frequently  <jiven  for  detailed  study  to  young  physicists  in 
order  to  train  tiicrn  in  the  art  of  raading  for  themselves  period- 
ical scientific  literature.  Certainly  no  better  piece  of  work 
could  be  used  for  the  purpose,  whether  one  considers  the 
intrinsic  importance  of  the  subject-matter,  the  keenness  of 
argument  and  the  logical  presentation  in  detail,  or  the  use  and 
design  of  apparatus  and  the  treatment  of  sources  of  error. 
The  main  objections  to  Cavendish's  work  are  those  he  himself 
pointed  out,  and  it  is  important  to  notice  that,  notwithstand- 
ing all  the  advance  in  the  refinement  and  manipulation  of 
apparatus  which  has  been  made  during  the  century  that  has 
elapsed  since  the  date  of  Cavendish's  experiment,  his  value  for 
the  mean  specific  gravity  of  the  earth,  5.448,  must  still  be  con- 
sidered one  of  the  most  reliable,  being  not  far  from  the  latest 
results  of  Poynting,  Konig  and  Richarz  and  Krigar-Menzel, 
Boys  and  Braun. 

Believing  that  we  in  America  devote  insufficient  time,  if 
any,  to  a  study  of  Newton's  great  work,  the  editor  has  thought 
it  well  to  incorporate  with  the  memoirs  on  the  experimental 
investigation  of  gravitational  attraction  the  statements  of  New- 
ton himself  concerning  that  subject. 

The  laws  of  gravitation  are  embodied  in  the  formula, 

mm' 

which  says  that  the  attraction  between  two  particles  of  matter 
is  directly  proportional  to  the  product  of  their  masses,  inverse- 
ly proportional  to  the  square  of  the  distance  between  them, 
and  independent  of  the  kind  of  matter  and  of  the  intervening 


'c«l  value  of  o     r "    '.    r^"''  ^''^^le  usTo  Z  .. *['•"•    ^"'='' 
"•action  beueon  ?    ""  P"'''"'^  >ve  mwt  1^^  ""^  """•«•- 

Mperimcts  vZ  f"     "'  '"  "^O'''-     Aa  the  ,„al!'"'™"=""'  ""at 

"""•'h's  crusr  «n  ,  ""'S^  mountain  ma^!  „  !"  '"""'s  of  an 
been  obseved  tm ''  ",'  ""«k»e«8,  a^da^  ^, '"?''""'<"  "-e 
""«'  'he  Su;;';  «;f  different  inr„mentr  » ".r"'""  I"" 
»nce  and  the  bo '"k  ,""  ^''''"'  "valance  tZ  ^'"'  P'"""" 
always  abo^^t  tlT  '"'"'™'  "■"'  ^et  the  1,  u  P""*!"'"™  bal- 
eonstitutin:,  ftM"""'  "^  <'''"  wT '"'^ '''''■'e  of  G  is 
h-s  aocordfn^,'"  '7'  P'-oo'of  Ne /ton's  la ^!  "Periments  as 

«"™ing  the  efrfh  f  ,  ^"'"*«<'  '"  "sine  the  ,'.r"''."'«  «<"'»•• 
*'th  the  vX^/"  "«  a  sphere,  the  vflne  o/"'.^"^''-  As- 
"^  ".e  eqnat!::  "'  '"^  •"«""  'Pacific  gravity^'o^'.V::;;,^;'^ 

Where  n  fg  thfi  o««  i  ^^ti^ 

the  earth     a„V  ^^'^'•^^'«»  d»e  to  ^raviti.       ^  t> 

«'"^  o    the  aboVr'^'^"^''^'  ^'^  '•«  quite  usull^  "^  J^^  ^*^^'°«  «' 

the  earth.     '^"'^  ^^Penmeuts  is  I  VdZ  L""''.  '^^^  *^« 

^^he  work  on  ^^'^''J'  ^^ 

- -'Strr  «»Serand'deTrd"   ~  "^  *"« 

P>-esen..d.     It  Z,u'  '""""""  of  the  Hroifc  t!  ,  ''^^«''»M  to 
«f  the  method,  wi  ^k"  T'"  """  "'ey  weTe  th.  n  "''^'''«  ''«'•« 
g'-avitational  attlf''  ""'"«  ''een  used  for  the  '^  ""'""  '"  ""> 
feet  instr,,ment''rT"  '  '''■'<'  "'tl.ongh    1    '"""'"'•^^ent  of 
merical  results  !""  ""'"'onrable  ifcal  o„n^v""'  "'  ''npef- 

vi  ^^®  '"^"^^''•s  as  seemed 


^.*ifei.„ 


itation  Con- 
l»e  law,  than 
en  from  the 
^al  formula, 
•fiirect,  for, 
show  that 
dies.    Such 
'he  nnmer- 
'"0  the  at- 
t  a  known 
•acter  that 
3d  in  such 
'ths  of  an 
loIl  of  the 
action  has 
»e  plumb- 
Jlum  bal- 
;ie  of  G  is 
i^nents  as 
iie  editor 
en.     As- 
onnected 
sarth.  A, 


PUEFACK 

necessary  to  prevent  the  reader  from  wasting  time  over  obscure 
and  iuaci^urate  passages,  and  to  suggest  material  for  collateral 
reading. 

An  effort  luis  been  made  to  present  along  with  the  memoirs 
I  brief  historical  account  of  the  various  modes  of  experiment 
used  fur  finding  the  mean  specific  gravity  of  the  earth,  and  a 
table  of  results  is  added.  As  the  literature  on  the  subject 
before  the  present  century  is  not  always  easily  obtainable,  the 
treatment  of  the  matter  for  that  period  is  given  in  compara- 
tively greater  detail.  Believing  that  a  bibliography  contain- 
ing every  important  reference  to  the  subject  is  an  essential 
feature  of  a  work  of  this  kind,  the  editor  has  endeavoured  to 
make  himself  familiar  with  the  whole  of  the  very  extensive 
literature  relating  to  it,  and  accordingly  is  fairly  confident 
that  no  important  memoir  has  escaped  his  observation.  From 
the  mass  of  material  ihus  collected  the  bibliography  given  at 
the  end  of  the  volume  has  been  compiled.  In  order  to  keep 
within  the  limits  of  space  assigned,  some  references  had  to 
be  omitted,  but  they  relate  mainly  to  recent  work,  and  it  h 
believed  that  they  contain  nothing  of  importance. 

No  effort  has  been  made  to  deal  with  the  mathematical  side 
of  the  subject;  accordingly  the  memoirs  of  Laplace,  Legendre, 
Ivory,  etc.,  which  deal  with  the  finding  of  the  mean  specific 
gravity  of  the  earth  by  means  of  analytical  methods  are  not 
referred  to;  but  it  is  hoped  that  all  the  more  important  ex- 
perimental  investigations  have  been  touched  upon. 


Bhtn  Mawr,  Ocicbfr,  1899. 


A.  Stanley  Mackenzie. 


?tt 


py  the 
eru  is 
Ives  to 
here 
n  two 
nt  of 
nper- 
nu- 
and 
lear- 
med 


I  If 


1^ 


HISTORY  OF  THE  SUBJECT 

BEFORE   THE 

APPEARANCE  OF  NEWTON'S  ''PRINCIPIA" 


Dk.  (cILHEUt's  contributions  to  the  spocultitions  on  i^ruvita- 
tion  jiro  lunon.i!;  the  most  iniportjint  of  the  etuiy  writinjj^s  on  tliat 
Kuhject,  ultlioiisjh  to  Ke}>k'r  also  must  eredit  he  ^iven  for  a 
deep  insight  into  its  nature;  tiie  hitter  announces  in  his  intro- 
duction to  tli^  Astronomid  Soiui,  published  in  1(!0!>,  his  belief 
iu  the  i)erfect  recdpnxMty  of  the  action  of  gravitation,  and  in  its 
application  to  the  whole  material  universe.  (Jilbert  was  led  by 
his  researches  on  magnetism  to  the  conclusion  that  the  force  of 
gravity  was  due  to  the  nnignetic  properties  of  the  earth;  and 
in  IGOO  announce<l  [1*,  I,  21]  iiis  opinion  that  bodies  when  re- 
moved to  a  great  distance  from  the  earth  would  gradually  lose 
their  motion  downwards.  The  earliest  proposals  we  find  for 
investigating  whether  such  changes  occur  in  ihe  force  of  gravity 
are  in  the  works  of  Francis  Bacon  Y'i,  Nov.  On).  II,  3G,  and 
Itist.  Ndt.  I,  153].  lie  maintained  that  this  force  decreased 
l)oth  inwards  and  outwards  from  the  surface  of  the  earth,  and 
suggested  experiments  to  test  his  views.  He  would  take  two 
clocks,  one  actuated  by  weights  and  the  other  by  the  compres- 
sion of  an  iron  spring,  and  regulate  them  so  that  they  would 
run  at  the  same  rate.  The  clock  actuated  bv  weights  was  then 
to  be  placed  at  the  top  of  some  high  steej)le,  and  at  the  bottom 
of  a  mine,  and  its  rate  at  each  place  compared  with  that  of  the 
other,  which  remained  at  the  surface.  There  is  no  record  of 
any  trial  of  the  experiment  at  that  time. 

After  the  founding  of  the  Royal  Society  of  London  a  stimu- 
lus was  given  to  experimenting  upon  this  as  upon  many  otiier 


*  The  inimbers  in  brackets  refer  to  the  Bibliography. 

1 


i» 


I  /( 


^^^M  01  its    0]v 
wore  uutTl  ^'''""^    ^^^'^^''t   and     '^       ^^^^''raneous 

-  «;-..i  to  the ««.;:.  ;L,:;  ii^ix  ""'"•■'r'  "^  •-»- 

ofthpm«H     1   ^     ^   Js  worth  repri.'tintr    "    ^."^    ^^e8tmlnster 
.  ,  '^  method  employed  in  .„p),  ^^'  "^^  ^'^ing  some  idp;* 

of  ^^"owledge  upon  ti  c         f  "t'tt'T?" '"'^'  ^^"^  ^^  theVt^L 
measured  fho  ^....fi     xr       '      "'  ^'card  havino- ;..  n  -^  ^ 


THE    L  A  W  S    OF    ( J  It  A  \'  I  T  A  T  1  O  N 


'V  Dr.  Power 
ubternuieous 
8    of   tlireud 
)oised.     The 
by  means  of 
nouth  of  the 
ce.*     Tliree 
'port  to  the 
iV^estminster 
g  some  idea 
of  the  state 
Newton  first 
n  1GG5  that 
nagine  that 
all  changes 
md  be  the 
irsuing  his 
le  sun  and 
ws,  it  was 
lie  inverse 
this  law  of 

in  deduc- 

estimate, 
nstead  of 
ion  of  the 
hereupon 
upon  the 

ect  by  a 
neantime 

ect  data 

t  of  the 
Prom 

fonnda- 

It: 

oncern- 
ved  be- 
iy  such 
•y  their 


nearer  or  farther  removal  from  that  siirfaoe  upwards.  To  this 
end  I  took  a  [)air  of  exact  scales  and  weights,  and  we!it  to  a 
convenient  place  upon  Westminster  Abbey,  where  was  a  per- 
pendicular height  above  the  leads  of  a  subjacent  l)uildiiig. 
which  by  measure  I  found  tiireescore  and  eleven  foot.  Here 
counterpoising  a  piece  of  iron  (wliicb  weighed  about  15  ouiu;es 
troy)  aiul  packthread  enough  to  reach  from  the  top  to  the  bot- 
tom, 1  found  the  counterpoise  to  be  of  troy- weight  seventoon 
ounces  and  thirty  grains.  Then  letting  down  the  iron  by  the 
thread,  till  it  almost  touched  the  subjacent  leads,  I  tried  what 
alteration  there  had  happened  to  its  weight,  and  fouiul,  that 
the  iron  preponderated  the  former  counterpoise  somewhat  more 
than  ten  grains.  Then  drawing  up  the  iron  and  thread  with 
all  the  diligence  possibly  I  could,  that  it  might  neither  get  nor 
lose  any  thing  by  touching  the  perpend icniar  wall,  1  found  by 
putting  the  iron  and  packthread  again  into  its  scale,  that  it 
kept  its  last  equilibrium;  and  therefore  concluded,  that  it  had 
not  received  any  sensible  difference  of  weight  from  its  nearness 
to  or  distance  from  the  earth.  I  repeated  the  trial  in  the  sunie 
place,  but  found,  that  it  had  not  altered  its  e(|uilibrium  (as  in 
the  first  trial)  neither  at  the  bottom,  nor  after  I  had  drawn  it 
up  again  ;  which  made  me  guess,  that  the  first  preponderating 
of  the  scale  was  fro!n  the  moisture  of  the  air,  or  the  like,  that 
had  stuck  to  the  string,  and  so  made  it  heavier.  In  pursuance 
of  this  experiment,  I  removed  to  another  place  of  the  Abbey, 
that  was  just  the  same  distance  from  the  ground,  that  the  for- 
mer was  from  the  leads;  and  upon  repeating  the  trial  there 
with  the  former  diligence,  I  found  not  any  sensible  alteration 
of  the  equilibrium,  either  before  or  after  I  had  drawn  it  up; 
which  farther  confirmed  me,  that  the  first  alteration  proceeded 
from  some  other  accident,  and  not  from  the  differing  gravity  of 
the  same  body. 

"I  think  therefore  it  were  very  desirable,  from  the  determi- 
nation of  Dr.  Power's  trials,  wherein  he  found  such  difference 
of  weight,  that  it  were  examined  by  such  as  have  opportunity, 
first,  what  difference  there  is  in  the  density  and  pressure  of 
the  air,  and  what  of  that  condensation  of  gravity  may  be  as- 
cribed to  the  differing  degrees  of  heat  and  cold  at  the  top 
and  bottom,  which  may  be  easily  tried  with  a  common  weather- 
glass and  a  sealed-up  thermometer  ;  for  the  thermometer  will 
shew  what  of  the  change  is  to  be  ascribed  to  heat  and  cold, 

8 


I  It 


!     ' 


i    I 


1  I 


Ml 


il, 


f 


;  ! 


,    !l 


MK  MO  I  IIS    ON 


Mini  tlic  wciUlicr  j^Miiss  will  show  tlic  (lincriTi*,'  condciiKHtion. 
Next,  for  t,li(!  knowing;,  wliotlior  this  ultcrutioii  of  «j;riivity  pro- 
cchmI  from  tho  doMsity  mid  ^'nivity  of  tho  iiiiihicnt  siir.  it  would 
l»o  requisite  to  niuke  use  of  some  vei-y  li^lit  ixxly,  exteiidiMl 
into  hiv^ii  dimensions,  sueli  jis  u  liirjje  ;j;lol)e  of  j^diiss  carefully 
stopt,  that  no  uir  may  get  in  or  out;  for  if  the  alteration 
proceeded  from  the  magnetical  attraction  of  the  parts  of  the 
earth,  the  hail  will  lose  l)ut  a  sixteenth  j)art  of  its  weight  (suj)- 
posing  a  lunjp  of  glass  held  tlie  same  pro))ortion,  that  l>r. 
Power  found  in  hrass) ;  hut  if  it  jiroceed  from  the  (lensity  of 
the  air,  it  may  lose  half,  or  perhaps  more.  Further,  it  wen; 
very  desireahh^,  that  th(i  current  of  the  air  in  that  })la(!e  were 
ohscrved,  as  Sir  Rohert  Moray  intimated  the  last  day.  Fourth- 
ly, I  think  it  were  worth  trial  to  counterpoise  a  light  and  heavy 
body  one  against  anotlier  above,  and  to  carry  down  the  scales 
and  them  to  the  bottom,  and  observe  what  happens.  Fifthly, 
it  were  dcsireablc,  that  trials  were  n»ade,  by  the  letting  down  of 
other  both  heavier  and  lighter  bodies,  as  lead,  quicksilver,  gold, 
stones,  wood,  liquors,  animal  substances,  and  the  like.  Sixthly, 
it  were  to  be  wished,  that  trial  were  made  how  that  gravitation 
docs  decrease  with  the  descent  of  the  body — that  is,  by  making 
trial,  how  much  the  body  grows  lighter  at  every  ten  or  twenty 
foot  distance.  These  trials,  if  accurately  made,  would  afford 
a  great  help  to  guess  at  the  cause  of  this  strange  phaenom- 
enon." 

Dr.  Power's  experiment  was  repeated  by  Dr.  Cotton,  and  an 
account  of  his  trials  was  given  to  the  Society  on  June  1,  1004 
[10,  vol.  1,  p.  433].  The  weight  was  ^  lb.,  and  the  length  of  the 
string  3(J  yards.     A  loss  in  weight  of  ^  oz.  was  found. 

On  September  1,  16G4  [G,  vol.  5,  p.  307],  we  find  a  reference 
to  some  experiments  made  at  St.  I*aul's  Cathedral  by  a  com- 
mittee of  the  Koyal  Society  consisting  of  Sir  R.  Moray,  Dr. 
Wilkins,  Dr.  Goddard,  Mr.  Palmer,  Mr.  Hill  and  Mr.  llooke. 
The  results  of  these  experiments  were  given  to  the  Society  on 
September  14,  1004  [10,  vol.  1,  p.  4GG]  ;  the  weight  was  15  lbs. 
troy,  the  string  about  200  ft.  long,  and  the  loss  of  weight  1 
drachm.  In  a  letter  to  Mr.  Boyle  [G,  vol.  5,  p.  53G],  dated 
September  15th,  Mr.  Ilooke  gives  more  details,  and  remarks 
that  the  balance  was  sensitive  enough  to  be  turned  by  a  few 
grains.  He  suggests  the  variation  of  the  density  of  the  air  as 
the  cause  of  the  loss  in  weight.     Boyle  [10,  vol.  1,  p.  470]  pro- 


Li! 


■K 


TIIK    LAWS    OK    (iKAVITATIOX 

posod    tliiit    II()()kt''s   snijixostioii   be  tested   by   miikin<;  tlio  hus- 
pended  \vei<;lit  of  u  l;iri;e  <;liiss  hull  loaded  with  menuiry. 

At  a  iiieetiii<if  of  the  lioyal  Society  on  Maieh  14,  Hid.").  Ilooke 
rejxnted  (10,  vol.  'i,  ]>.  <»(l,  and  i>,  vol.  .">.  p.  ")44|  that  ho  had 
tried  Dr.  Power's  (vxperimeni  at  some  wells  near  Mpsom  and 
had  found  no  loss  in  wei<jflit.  Similar  experirjients  wen;  made 
hv  Ilooke  at  Hanstead  Downs,  in  Surrey,  and  reporteil  on 
Mar(d)  '-il,  Hidfl  (10,  vol.  '■!,  j).  <»'.>,  and  (J,  vol.  a,  pp.  ;{">ri  and 
r)4<!l.  The  strini,'  was  X\()  ft.  lon^.  and  the  halanee  sensitive 
to  a  i^raiji,  yet  a  pound  shewed  no  <dianj,'e  in  weii.dit  when  sus- 
jteiided  at  the  bottom  of  the  well.  He  concludes  that  the 
j)ower  of  «;ravity  cannot  bo  niaj^netical,  as  (lilbort  had  sup- 
posed, lie  says:  "  Hut  in  truth  upon  the  c(msidon:tion  of  the 
nature  of  the  theory,  we  may  find,  that  supposing:;  it  true,  that 
all  tlie  constituent  parts  of  the  earth  had  a  mMi,MU'tical  power, 
the  decrease  of  <]^ravity  would  be  almost  a  hundred  tinu's  less 
than  a  jj^rain  to  a  })ound,  at  as  fjreat  a  depth  as  lifty  fathom  : 
for  if  we  consider  the  ])roportion  of  the  parts  of  tlu^  earth 
placed  uj)on  one  side  beneath  the  stone,  with  the  parts  on  tho 
other  side  ai)ove  it,  we  may  fiiul  the  disproportion  greater. 
Unless  we  sup|)oso  the  majjjnetism  of  the  i)arts  to  act  but  at  a 
very  little  distaiuH'.  which  I  think  the  experiments  made  in  tho 
Abbey  and  St.  Paul's  will  iu)t  allow  of.  If  therefore  there  be 
any  su(di  inequality  of  j]fr;ivity,  we  must  have*  some  ways  of 
trial  much  more  accuirato  than  this  of  scales,  of  whi(di  I  shall 
propound  two  sorts,"  etc.  It  is  interestiujr?  to  notice  that  the 
considerati(ms  upon  which  ho  makes  his  (;omputations  are  prac- 
tically those  used  by  Airy  in  his  Harton  Colliery  exj)eriment. 

On  December  7.  1081  [10,  vol.  4,  p.  110],  Ilooke  i)roduced 
before  tho  society  two  pendulum-clocks  adjusted  to  run  at  tho 
.same  rate,  lie  proj)()sod  to  put  one  at  tho  toj)  and  the  other 
at  tho  bottom  of  tho  monument  on  Fish  Str(;('t  Hill,  and  ob- 
serve wliether  they  would  keep  to<^ether.  No  notitui  of  his 
having  tried  the  experiment  has  been  fouiul.  Thh  is  the 
method  proposed  by  Ha(!on  and  used  by  Bouguer  and  many 
others. 

In  108:2,  Ilooke  read  before  the  Royal  Society  *'  A  Discourse 
of  the  Nature  of  Comets"  [4,  pp.  I4H-11I1],  in  whicli  he  gives 
his  ideas  on  the  subject  of  gravity  (})articularly  on  pages  170- 
18;}).      Ho   considers  gravity  to  be  a  universal   principle,  in 
herent  in  all  matter,  pro})agated  by  the  same  medium  as  that 


H 


i  ; 


f     t 


MKMOIHS    ON    TIIK    LAWS    OK    (JKAVITATION 

l)y  nietuiH  of  wlii(;li  li^Hit  is  conveyed,  witli  unimaginable  celer- 
ity, to  indefinitely  gri^at  distances,  and  with  a  power  varying 
witjj  the  distan(re.  He  sums  up  his  conceptions  on  gravitation 
in  nine  i)roj)ositions,  wliich  are  of  great  interest,  in  that  they 
inchnhi  many  of  the  conceptions  of  Newton  on  tliis  subject, 
and  yet  were  published  four  years  before  the  Prinripia  ap- 
peared. 

6 


ITATION 

rinable  celer- 
)\ver  varying 
1  gravitation 
in  tliat  they 
this  subject, 
rincipia  ap- 


PHILOSOPHIAE  Ni^TURALLS   PRINCIPIA 

MATHEMATICA 

1*^  Edition,  Iymdo,i,  1687.     2d  Kditiou,  ('a,„t,nd,,f,  1713  (CoteH'  Edition) 
ddMUton,  London,  1736  {Pe>n/>ertoii'fi  Addition) 

AND 

DE  MUNDl  SYSTEMATE 

Ijmdon,  1727 


BY 


SIR  ISAAC    NEWTON 


{Extracts  taken  from  Dmrn's  Edition  of  Motte's  translation 
3  volumes,  J^/ulon,  1803) 


CONTENTS 

PAdK 

On  the  attraction  of  spfnirs \) 

Lain  of  the  diKtance 9 

Jmw  of  the  inaHHi'H 13 

Variation  of  (j rarity  on  the  earth' a  mirfaee 14 

All  attraction  in  mutual l.") 

Methods  of  xhowing  the  attraction  between  terrextrial  Itodien 17 

Proof  of  itH  ej'intcnce 17 

Similar  discuxaion  for  the  cane  of  celestial  Itodien IH 

Final  statements  concerning  tfie  laws  of  gravitation 19 

8 


i! 


^ 


THE    MATHEMATICAL    PUIN(.MPLES    OF 
NATURAL   PHJLOSOFHY 

AND 

SYSTEM    OE    THE    WORLD 

HY 

SIR    ISAA(J    NKWTON 


|i(K>K    I.       I*K(HM)SITI()N     LXXIV.       TllKollKM    XXXIV. 

Tin'  same  f/tinf/s  .stf/i/tnscd  (if  to  tho  sovonil  points  of  :i  j^iven 
8|)li(!r(i  tluM'e  toiul  c(|iiJil  cciitripotal  forces  (lecroasiriij  in  u  dii- 
plicjito  ratio  of  tlio  distiiiici'S  from  tiio  points),  /  sat/,  llnd  a  ntr- 
pHsrle  silnafe  without  tlic  s/i/iere  is  ((ttntrtrd  witli  <(  /nrm  rcri/}- 
rocftllj/  proportional  to  tlw  sf/uarc  of  its  distance  from  the 
centre. 


Book  I.     Puoposition^  LXXV.     Tiikokkm  XXXV. 

//■  to  the  several  points  of  a  t/iren  sphere  there  tend  ef/aal  cen- 
tripetal forces  decreasing  in  a  dnplic((te  ratio  (f  the  dist(tnres 
from  tlie  points  ;  I  sat/,  that  another  similar  sphere  will  f/e  altrart- 
ed  1)11  it  with  a  force  recijrrocallf/  projwrtional  to  the  square  of  the 
distance  of  the  centres. 

For  the  attraction  of  every  particle  is  reciprocally  as  tlie 
square  of  its  distance  from  the  centre  of  the  attracting  spiiere 
(by  prop.  74),  and  is  therefore  the  same  as  if  that  whole  at- 
tracting force  issued  from  one  single  corpuscle  placed  in  the 
centre  of  this  sphere.  But  this  attraction  is  as  great  as  on  the 
other  hand  the  attraction  of  the  same  corpuscle  would  be,  if 
that  were  itself  attracted  by  the  several  particles  of  the  attract- 
ed sphere  with  the  same  force  with  which  they  are  attracted  by 

9 


MKNKHUS    (jN 

It.  Rut  timt  iittnu'tio!!  of  tlio  corpiisclo  would  he  (hy  prop.  T4) 
n'ci|>r(t("illy  iiroporiioiuil  to  llir  s(|iiar('  of  its  (listiiii(«(  fiotn  tlio 
(■('Mtic  of  the  splicrc  ;  I  licrrforc  llic  jittnictioii  of  tlu^  spluTc, 
iMjiiiil  llicirto,  is  also  in  tlic  siiiiic  ratio.  (^.   K.   D. 

Cor.  I.  The  atliiU'lions  of  s|»li('i«'s  ttnvards  otiior  li(nnot;oii('- 
oiiH  spheres  are  as  tlie  attnietiiiiT  sphcMcs  applied  to  tin?  squares 
fd"  the  distatiees  <d'  their  centres  from  the  centres  of  those 
\vlii(di  they  attract. 

Cor.  '^.  'IMie  cas(f  is  tlio  same  wiien  the  attracted  sphere  does 
also  attract.  Kor  the  several  points  of  the  one  attract  the  sev- 
«'ral  points  of  the  other  with  the  sainc;  force  with  whi<di  they 
themselves  are  attracted  hy  the  (tthers  a;;ain  ;  and  therefore 
since  in  all  attractions  (hy  law  11)  the  attracted  and  attractin.i^ 
point  are  both  c(|indly  acte(l  on,  tlu^  force  will  he  doubled  by 
their  mntnal  attractions,  the  proportions  retnainini;. 

[  I'ro/m.si/ioH  /y.\'.\'  r/.  /wnrcs  I  he  sttniP  Ihiiuj  fur  sfthprvs  tHftrlr 
up  of  honnif/t'iu'oHs  ronrrnfrir  liti/cr.s.  | 

Rook  III.     Rkoi'osition   \'.     Tiikokkm   V.     Sciioijim. 

The  force  which  retains  the  celestial  bodies  in  their  orbits 
has  been  hitherto  called  centripetal  force;  but  it  being  now 
made  plain  that  it  can  bo  no  other  than  a  j;ravitatin<(  force, 
we  shall  hereafter  call  it  gravity.  For  tlie  cause  of  that  cen- 
tripetal force  whi<di  retains  the  moon  in  its  orbit  will  extend 
itself  to  all  the  planets. 


' 


Rook  III.     Proposition  VI.     TiiKoi{K>r  VI. 

T/inf  all  Imlivx  f/niri/dfc  fofrnrds  ct^'cri/  phiucf  ;  and  that  the 
wcighfs  of  bodies  tomirds  (un/  the  same  planet,  at  er/fial  distances 
from  the  centre  of  the  planet,  are  proportional  to  the  quantities 
of  matter  irhich  Ihej/  severnllji  contain. 

It  has  been,  now  of  a  long  time,  observed  by  others,  tbat  all 
sorts  of  beavy  bodies  (allowance  being  made  for  the  inequality 
of  retardation  wbicb  they  sutler  from  a  small  power  of  resist- 
ance in  tbc  air)  descend  to  the  eartb  from  equal  heifjhts  in 
equal  times  ;  and  tbat  equality  of  times  we  may  distinguish  to 
a  great  accuracy,  by  the  help  of  pendulums.  I  tried  the  thing 
in  gold,  silver,  lead,  glass,  sand,  common  salt,  wood,  water, 
and  wheat.     I  provided  two  wooden  boxes,  round  and  equal ; 

10 


5 


•<»  (l>y  prop.  74) 
iitici!  fi-oni  tlio 
>f  the  splu'ic, 

(^  K.  [). 

I*^'!'    llOI||()(r(.||0. 
to  tll(!  .Sfj||lirp.S 

itrt'S    of    those 

<l  sphere  does 
tract  tho  Hov- 
li  which  they 
m(l  therefore 
ml  JiMnietiriir 

I'  <loiihle(l  hy 
\<^ 

sp/ierr.s  tnade 


tlieir  orbits 
t  l)eiii^'  now 

iitiii;.r  force, 
of  that  (H'ji- 

will  exteiiil 


VI. 

nd  that  the 
I  flis/anrcft 
qKanlities 

••s,  that  Jill 
inequality 

*  of  res  is  t- 
hcifjhts  in 
n^uish  to 
the  thing 

5d,  water, 

id  equal ; 


TIIK    LAWS    OF    (Mt.WITATloN 

I  filled  the  one  with  wood,  juid  suspended  ;in  e(|n;d  weiirlit  of 
^'old  (as  exactly  as  I  coidd)  in  the  I'cntre  of  oscilhUion  of  the 
other.  'V\w  boxes  hMnirini;  bv  equal  threads  of  1  I  feet  nnide  a 
couple  of  pendidunis  perfectly  e(|ual  in  weii^Hit  and  lii^ure,  and 
erpudly  receivin*?  the  resistance  of  the  air.  And.  placiuij  the 
on»?  by  the  other,  I  observed  them  to  play  tnijether  fcu'wards 
atid  ba(  kwards,  for  a  loui;  time,  with  eqind  vil)rations.  .  .  . 
and  the  like  happened  in  tht!  other  bodies.  lU'  these  experi- 
ments, in  boilii's  of  the  satne  weiirlit,  I  could  immifestly  hav(» 
discovere(l  a  dilTerence  of  nuitter  less  than  tin'  thousandth  ]>art 
of  the  whole,  had  any  smdi  been.  Itiit,  without  all  doubt,  tho 
luiture  of  gravity  towards  the  planets  is  the  sanu'  as  towards 
the  earth.  .  .  .  iMoreover,  since  the  satellites  of  .lupiter  per- 
form their  revolutions  in  times  which  observe  the  sesfpiiplicate 
proportion  of  their  distances  from  .lujtiter's  centre,  their  acc»d- 
erativc  gravities  towards  .lupiter  will  be  reciprocally  as  tin? 
s(|uares  of  their distaiu.'es  from  .lupiter's  centre — that  is,  e(pial 
at  equal  distances.  Ami,  thertd'ore,  these  satellites,  if  sup- 
posed to  fall  fofiutrds  .In/tilrr  from  e(|ual  luugbts,  would  describe 
equal  spaces  in  eipuil  times,  in  lik(>  manner  as  heavy  bodies  do 
on  our  earth.  .  .  .  If,  at  e(|ual  distanc^es  from  the  sun.  any  sat- 
ellite, in  proportion  to  the  <|uantity  of  its  nuitter,  did  gravitate 
towards  the  sun  with  a  force  greater  than  .lupiter  in  propor- 
tion to  his,  according  to  any  given  proportion,  suppose  of  d  to 
(' ;  then  the  distance  between  the  centres  of  the  sun  and  of  the 
satellite's  orbit  would  be  always  greater  than  the  distance  be- 
tween the  centres  of  the  sun  and  (»f  .lupiter  nearly  in  the  sid»- 
duplieate  of  that  proportion  ;  as  by  some  computations  I  have 
found.  And  if  the  satellite  did  gravitate  towards  the  sun 
with  a  force,  lesser  in  the  ])ro])ortion  of  e  to  (L  the  distance  of 
Llie  centre  of  the  satellite's  orbit  from  the  sun  would  be  less 
than  the  distance  of  the  centre  of  .lupiter  from  the  sun  in  the 
subduplicate  of  the  same  proportion.  Therefore  if,  at  equal 
distances  from  the  sun,  the  accelerative  gravity  of  any  satell- 
ite towards  the  sun  were  greater  or  less  than  the  acu-elerative 
gravity  of  .Jupiter  towards  the  sun  but  by  oiu'  ^^}^^^^  part  of  the 
whole  gravity,  the  distance  of  the  centre  of  the  satellite's  orbit' 
from  the  sun  would  be  trreater  or  less  than  th(»  distar 


o 


u- 


f  J 

piter  from  the  sun  by  one  ^^j,„  part  of  the  whole  distance — 
that  is,  by  a  fifth  part  of  the  distance  of  the  utmost  satellite 
from  the  centre  of  Jupiter  ;  an  ; .  lentricity  of  the  orbit  which 

11 


.Avxy.ij:'!^. 


i    !!r 


'ii: 


'1 


Ml 

Ml 


« 


1 

I'  : 


'Ii 
i  i| 


mi:  Mo  I  Its    ON 

would  hn  vory  sfMHihlc  hiil  llir  «>il)it«  of  tlic  Hiitdlitoa  ari^ 
<'<)ti('(Mi( I'ic  tn  Jiipilri',  iiiid  t licrcfort;  tlu;  iKMUiici'iil ivc  ^raviiics 
of  .hipitcr,  iiml  of  .ill  it.s  Hiitt'lliti^s  towiinU  the  sum,  an;  iMpiul 
luiKni^  tliciiisclvt's.   .   .   . 

Hut  fiirtlicr;  tlio  wcij^lits  of  nil  tlu^  parts  of  every  planot 
to\var(ls  any  other  planet  art!  on(»  to  another  as  the  malter  in 
the  several  parts;  for  if  sonie  parts  did  ^M'avitate  more,  others 
less,  th;in  for  tln^  (pnintity  of  their  matter,  then  tln^  whoh* 
planet,  aecordinij  to  tlu^  sort  (d'  parts  with  whi<di  it  most 
aitonnds,  would  ;^M'avitato  mori^  or  less  than  in  proportion  to 
the  «(uantity  of  matter  in  the  whol(\  Nor  is  it  of  any  moment 
whether  these  parts  are  extermil  (U-  internal  :  for  if.  for  exam- 
ple, W(^  should  ima((ine  th(^  terrestrial  hodies  with  ns  to  ho 
raised  up  to  the  (U'l)  of  the  moon,  to  he  lh(>r(f  compared  with 
its  hody;  if  the  wcULjhts  of  suidi  hodies  were  to  the  wei<fhts  of 
the  external  parts  of  the  moon  as  the  (|iuintities  of  malter  in 
the  one  and  in  tlui  otluM*  respectively  ;  but  to  the  weij^hts  of 
the  inttuMial  parts  in  a  <,'r(!at(M'  or  less  proportion,  then  lik(!wiso 
the  wei.ujhts  of  those  hodi(»s  would  he  to  the  wei^dit  of  the 
whole  moon  in  a  j^reater  or  less  proportion  ;  uj^ainst  what  we 
have  slunved  ahove. 

Cor.  1.  IFence  the  weiifhia  of  hodies  do  not  de|)en<l  upon 
their  forms  and  texturcis  ;  for  if  the  weiufhts  coidd  ho  altered 
with  the  forms,  tlu^y  would  he  j:jreat(!r  or  less,  a(M'ordinsf  to  the 
variety  of  forins,  ine(|ual  matter;  altogether  a^ifainst  experience. 

Cor.  )i.  Universally,  all  hodies  ahout  the  earth  i^ravitate 
towards  the  earth  ;  ami  the  weijj^hts  of  all,  at  (Mpuil  distances 
from  the  earth's  centre,  are  as  the  (juantities  of  matter  whicdi 
they  severally  contain.  This  is  tlie  (|uality  of  all  bodies  within 
the  reach  of  our  (ixp(M-iments  ;  and  therefore  (by  rule  '.i)  to  bo 
artirmed  of  all  bodies  whatsoever.  .  .  . 

Cor.  T).  The  power  of  ujravity  is  of  a  different  nature  from  the 
power  of  magnetism  ;  for  the  mairnetic  attraction  is  not  as  the 
matter  attracted.  Some  bodies  are  attracted  more  by  the 
majjjnet  ;  others  less  ;  most  bodies  not  at  all.  The  power  of 
maijfiu'tism  in  one  and  the  same  body  may  bo  increased  and 
diminisluwl  ;  and  is  sometimes  far  stroni^er,  for  the  quantity  of 
matter,  than  the  power  of  Ljravity  ;  aiul  in  receding  from  the 
magnet  decreases  iu)t  in  the  duplicate  but  almost  in  the  tri- 
plicate proportion  of  the  distance,  as  nearly  as  I  could  judge 
from  some  rutle  observations. 

12 


Til  K     LAWS    ()|«     <i  K  \V  n ^\TlnN 


sllt('llit^^^  aro 
iv(^  ^ruviiit's 
III,  uro  l'(|iihI 

fvory  planol 
\o  iiijith'i-  ill 
Miorc.  oiln'is 
I  llio  wliolo 
i<*li  it  iiiost 
roportioii  to 
iiiiy  iiiornt'iit 
r.  for  oxiiin- 

tll      IIM     t«»     l)l> 

n pared  with 
<'  wci^'lits  of 
>f  Mijitlcr  ill 
>  \v('i«^r|its  of 
UMi  likewise' 
'i^'iit  of  tlio 
ii.sL  wliat  we 

I'pOIld    llJ)OM 

I  1)0  jiitei'i'd 

ding  to  tiio 

''xpcrit'iioe. 

I    ijfi'iivitiite 

I  distiiiices 

ttiT  wiiicli 

lies  Willi  in 

lo  ;})  to  bo 

from  the 
Inot  us  the 
re    by   tlie 

l)ower  of 
hiisod  jiml 
Quantity  of 

I  from  the 
li  tile  tri- 

bld  judge 


liooK   III.     I'liui'osnioN   VII.     'riiroitiiM   \'ll. 

y/nt/  f/irrr  is  n  fintnr  nf  f/rttrifif  fr/it/iiti/  In  if//  Intiliis,  inn- 
fHiffinmt/  In  lliv  si'iri'ol  iiiiiinl il iis  nf'  nuillvi'  ivlnrh  llirif  i ntiliNH. 

'riiiit  all  llic  plaiu'ls  iiiiitiially  ;;i'avitat«'  one  towards  aiiotlicr, 
we  have  prove*!  before;  as  well  as  that  the  fon-e  ol'  ;;iavity 
towards  every  ""e  of  IIh'Mi.  coiisidenMl  apart,  is  reeiproeally  as 
the  S(piar(!  <d'  (he  distanet^  of  plact's  froiii  the  centre  of  the 
plam't.  And  thence  (by  prop,  till,  book  I,  and  its  cor(dlaries) 
it  I'olhtws,  that  tli»!  gravity  lending  towards  all  the  planets  is 
proportional  to  the  matter  wliiidi  tlu>y  contain. 

>loreovcr,  since  all  the  parts  of  any  planet  A  gra\itate  to- 
wards any  other  planet  H;  and  tlio  gravity  of  every  part  is  to 
the  gravity  of  the  whole  as  tiie  matter  of  the  part  to  tlu^  matter 
of  the  whole  ;  and  (by  law  :{)  to  every  ai^tion  corresponds  an 
e(pial  reaction  ;  theiud'oro  tlu!  planet  B  will,  on  the  other  hand, 
gravitate  towards  all  the  parts  of  the  planet  A  ;  and  its  gravity 
towards  any  one  jiart  will  be  to  the  gravity  towanis  the  wlioh^ 
as  the  matter  of  the  part  to  the  matter  of  tlu^  whole,     (l.  K.  I). 

Cor.  1.  Therid'ore  the  force  of  gravity  towards  any  whole 
planet  arises  from,  and  is  (jomponiided  of,  the  fonies  of  gravity 
towards  all  its  parts.  Magnetic  and  ele(!tri(!  attractions  alTord 
us  examples  of  this  ;  for  all  attraction  towards  the  wlioh;  arises 
from  the  attractions  towards  the  several  parts.  The  thing  may 
be  easily  understood  in  gravity,  if  we  consider  a  greater  planet 
us  formed  of  a  number  of  lesser  i)lanets  meeting  together  in 
OIK!  globe  ;  for  ln'tirc  it  wotdd  appear  llial  the  force  of  the  whole 
must  arise  from  the  lorces  of  the  component  parts.  If  it  is 
objected  that,  according  to  this  law,  all  bodies  with  us  must 
mutually  gravitate  one  towards  another,  I  answer,  that  since 
the  gravitation  towards  these  bodies  is  to  the  gravitation  to- 
wards the  whole  earth  as  these  ])odie8  are  to  the  whole  earth, 
the  gravitation  towards  them  must  be  far  less  than  to  fall  under 
the  observation  of  our  senses. 

Cor.  )i,.  The  force  of  gravity  towards  the  several  equal  par- 
ticles of  any  body  is  reciprocally  as  the  sepiare  of  the  distance 
of  j)lace8  from  the  particles  ;  as  appears  from  cor.  3,  prop.  74, 
book  I. 

[  Under proposil ion  A'' occurs  the  fol/owiiuj  important  passage:] 
However  the  planets  have  been  formed  while  they  were  yet 
in  fluid  masses,  all  the  heavier  matter  subsided  to  the  centre. 

13 


!:,Vl 


•ir^ 


;  !(! 


H 


lii 


M  KM  O  I  IIS    ON 

Since,  tlioreforo,  the  oomrium  matter  of  our  earth  on  the  8nr- 
fjice  thenfof  is  ubout  twUre  hh  licavy  us  water,  and  a  little  lower, 
in  niinos,  is  found  about  throe,  or  four,  or  even  fivo  times  more 
heavy,  it  is  j)r()bablo  that  tlie  quantity  of  the  whole  matter  of 
the  cartii  may  be  live  or  six  times  irreater  tlian  if  it  consisted 
all  of  water.* 

[l^ndi'r  propnsifioits  XVI 1 1,  and  XIX.,  Newton  proves  that 
the  axes  of'  the  planets  are  less  th((n  the  diamrters  drawn  perpen- 
diridar  to  the  axes.  He  shows  how  centrifugal  force  acts  in 
deterniininf/  the  fornt  of  the  earth,  and  dificnsses  the  nieasurenients 
of  terrestrial  arcs  k'nown.  at  that  time;  he  deduces  therefrom  that 
fjraritt/  will  tte  la.  rned  at  the  et/?(ator  hj/  ggj  (f  itself  and  that 
the  earth  will  be  higher  at  tlie  erjuator  titan  at  the  poles  bg  17.1 
miles. ^ 

Book   III.     Proposition   XX.     Pkobij:m  IV. 

To  fnd  and  compare  toget/ier  the  weights  of  bodies  in  the  dif- 
ferent regions  of  our  earth. 

Because  the  weights  of  the  unequal  legs  of  the  eaiuil  of  water 
ACQy/rrt  are  equal ;   and  the  weights  of  the  parts  proportional 

to  the  whole  legs,  and  alike  situated 
in  them,  are  one  to  another  as  the 
weights  of  the  wholes,  and  therefore 
equal  betwixt  themselves;  theweights 
of  equal  parts,  and  alike  situated  in 
Q  the  legs,  will  be  reciprocally  as  the 
legs — that  is,  reciprocally  as  230  to 
220.  And  the  case  is  the  same  in 
all  homogeneous  equal  bodies  alike 
situated  in  the  legs  of  the  canal. 
Their  weights  are  reciprocally  as  the 
legs — that  is,  reciprocally  as  the  dis- 
tances of  the  bodies  'rom  the  centre  of  the  earth.  Therefoie, 
if  the  bodies  are  situated  in  the  uppermost  parts  of  the  canals, 
or  on  the  surface  of  the  earth,  their  weights  will  be  one  to  an- 
other reciprocally  as  their  distances  from  the  centre.  And,  by 
the  same  argument,  the  weights  in  all  other  places  round  the 
whole  surface  of  the  earth  are  reciprocally  as  the  distances  of 

*[  Tliu  Wiia  (I  wonderfnlli/  good  f/ueits  on  Newton' a  p(irt,unc£  tlie  best  of  tlie 
later  deter  mi  nations  give  about  5.5  for  tlie  nican  specific  gravity  of  tlie  eartli.^ 

14 


»    I 


4111 


rth  on  the  sur- 
1  a  little  lower, 
fivu  times  more 
■'liole  matter  of 
if  it  consisted 

hn  proves  that 
drawn  perpen- 

force  acta  in 
"meusuremcnls 

thcrefnm  that 
tse/f,  and  that 
<e  poles  by  17.1 

M    IV. 

iies  in  the  dif- 

oanal  of  water 

}  proportional 

alike  situated 

notlier  as  the 

and  therefore 

!s;  the  weights 

:e  situated  in 

•ocaliy  as  the 

ily  as  230  to 

tlie  same  in 

hodies  alike 

the   canal. 

ocaliy  as  the 

y  as  the  dis- 

Therefoie, 

the  canals, 

e  one  to  an- 

And,  by 

s  round  the 

distances  of 

tlie  best  of  the 
of  the  earth.} 


THE    LAWS    OF    (J  K  A  V  I  T A  T  1  ()  N 

the  places  from  the  centre;  and,  thoroforo,  in  the  hypothesis 
of  the  earth's  being  a  spheroid,  are  given  in  projjortion. 

[Xewton  then  states  that  ''the  lengths  of  prndtfluins  viln-atinrj 
in  equal  times  are  as  the  forces  of  (jrartti/";  he  ennnierates  the 
r.rper intents  on  the  periods  of  pendnhuns  made  at  differeut  parts 
'if  the  eart/t's  surfare,  and  tests  his  roNtlusions. 

The  following  re>narks  appear  on  pp.  'ii)-l7}  of  .]fofte's  transla- 
fion  of  the  'Ule  Mnndi  Systemate,''  wherein  Newton,  after  a 
reference  to  his  pendulum  experiments,  yiven  on  p.  li  of  this 
volume,  says ;] 

Since  the  action  of  the  centripetal  force  upon  the  bodies  at- 
tracted is,  at  equal  distances,  proportional  to  the  quantities  of 
matter  in  those  bodies,  reason  requires  that  it  should  be  also 
proportional  to  the  quantity  of  matter  in  the  body  attracting. 

For  all  action  is  mutual,  and  (by  the  third  law  of  motion) 
makes  the  bodies  tnutually  to  approach  one  to  the  other,  and 
therefore  must  be  the  same  in  both  bodies.  It  is  true  that  we 
may  consider  one  body  as  attracting,  another  as  attracted;  but 
this  distinction  is  more  mathetnatical  than  natural.  The  at- 
traction is  really  common  of  either  to  other,  and  therefore  of 
the  same  kind  in  both. 

And  hence  it  is  that  the  attractive  force  is  found  in  both. 
The  sun  attracts  Jupiter  and  the  other  {)lanets ;  Jupiter  at- 
tracts its  satellites  ;  and,  for  the  same  ri^ason,  the  satellites  act 
as  well  one  upon  another  as  upon  Jupi  er,  and  all  the  planets 
mutually  one  upon  another. 

And  though  the  mutual  actions  of  two  planets  may  be  dis- 
tinguished and  considered  as  two,  by  which  each  attracts  the 
other,  yet,  as  those  actions  are  intermediate,  they  do  not  make 
but  one  operation  between  two  terms.  Two  bodies  may  be 
mutually  attracted  each  to  the  other  by  the  contraction  of  a 
cord  interposed.  There  is  a  double  cause  of  action,  to  wit,  the 
disposition  of  both  bodies,  as  well  as  a  double  action  in  so  far 
as  the  action  is  considered  as  upon  two  bodies  ;  but  as  betwixt 
two  bodies  it  is  but  one  single  one.  It  is  not  one  action  by 
which  the  sun  attracts  Jupiter,  and  another  by  whi(;h  Jupiter 
attracts  the  sun  ;  but  it  is  one  action  by  which  the  sun  and 
Jupiter  mutually  endeavour  to  approach  each  the  other.  By 
the  action  with  which  the  sun  attracts  Jupiter,  Jupiter  and 
the  sun  endeavour  to  come  nearer  together  (by  the  third  law  of 
motion);  and  by  the  action  with  which  Jii[»iter  attracts  the 

15 


I! 


|ill!. 


if 


I     f 


■:'ii| 


i::i 


'    !i    ' 


Uli'i 


!;'! 


I  I 


I  i 


il 


!  i 


iiili 


t;  i 


!   II. 


H 


MKMOIUS    UN 

sun,  likewise  Jii])itor  and  tlie  sun  endoavour  to  oomo  nearer  to- 
gether. Hut  the  sun  is  not  attracted  towards  Jupiter  by  a 
twofohl  action,  nor  Jui)iter  hy  a  twofohl  acliion  towards  tlie 
sun  ;  but  it  is  one  siugk;  intermediate  action,  by  whicjj  both 
a})proacli  nearer  togetlier. 

Tiuis  iron  draws  tlio  loadstone  as  well  as  the  loadstone 
draws  the  iron  ;  for  all  iron  in  tiie  neighbourhood  of  tlie  load- 
stone draws  otluM*  iron,  liut  the  action  betwixt  the  loadstone 
and  iron  is  single,  and  is  considered  as  single  by  tlie  philoso- 
])liers.  The  action  of  iron  upon  the  loadstone  is,  indeed,  the 
action  of  the  loadstone  betwixt  itself  and  the  iron,  by  which 
both  endeavour  to  come  nearer  together;  and  so  it  manifestly 
appears,  for  if  you  remove  the  loadstone  the  whole  force  of  the 
iron  almost  ceases. 

In  this  sense  it  is  that  we  are  to  conceive  one  single  action 
to  be  exerted  betwixt  two  planets,  arising  from  the  conspiring 
natures  of  both;  and  this  action  standing  in  the  same  relation 
to  both,  if  it  is  proiiortional  to  the  (juantity  of  matter  in  the 
one,  it  will  be  also  proportional  to  the  quantity  of  matter  in 
the  other. 

Perhaps  it  may  be  objected  that,  according  to  this  phil- 
osophy (prop.  74,  book  1),  all  bodies  should  mutually  attract 
one  another,  contrary  to  rlie  evidence  of  experiments  in  ter- 
restrial bodies;  but  I  answer  that  the  experiments  in  terres- 
trial bodies  come  to  no  account ;  for  the  attraction  of  homo- 
geneous spheres  near  their  surfaces  are  (by  prop.  72,  book 
i)  as  their  diameters.  Whence  a  sphere  of  one  foot  in  diam- 
eter, and  of  a  like  nature  to  the  earth,  would  attract  a  small 
body  placed  near  its  surface  with  a  force  20,000,000  *  times  less 
than  the  earth  would  do  if  placed  near  its  surface  ;  but  so 
small  a  force  could  produce  no  sensible  effect.  If  two  such 
spheres  were  distant  but  by  one-quarter  of  an  inch,  they  would 
not,  even  in  spaces  void  of  resistance,  come  together  by  the 
force  of  their  mutual  attraction  in  less  than  a  month's  time;f 

*\Tf  the  sphere  in  one  foot  in  diameter,  this  number  should  be  40,000,000, 
since  the  dinmeter  of  the  earth  is  about  40.000,000/<.  But  perhaps  Newton 
intended  to  say  a  sphere  of  one  foot  in  radius.] 

f  [The  time  is  very  much  lefi.  On  the  assumption  that  each  of  the  spheres  is 
one  foot  in  diameter,  Poi/ntinf/  (185,  p.  10)  finds  tite  time  to  Ite  about  320  sec- 
onds. If,  however,  we  take  one  foot  as  the  radius  of  each  spliere,  Todhunter 
(J 40,  vol.  1,  p.  401)  >ih(noH  that  the  time  is  less  than  250  seconds.] 

16 


II 


THE    LAWS    OK    GRAVITATION 


ind  loss  splieres  will  come  together  ut  a  rate  yet  slower,  viz., 
Ill  the  proportion  of  their  diameters.  Nay,  whole  mountains 
|ivill  not  be  suttlcient  to  produce  any  sensible  effect.  A  moun- 
tain of  an  hemispherical  figure,  three  miles  high  and  six  broad, 

kvill   not,    by   its    attraction,  draw    the    pendulum    two  min- 

^ites  *  out  of  the  true  perpendicular;  and  it  is  only  in  the 

freat  bodies  of  the   planets  that  these  fo;*ces  are  to  be  per- 

•eived,  unless  we  may  reason  about  suuiller  bodies  in  manner 

followiug.f 

Let  A  BCD  represent  the  globe  of 
the  earth  cut  by  any  plane,  AC,  into 

[two  parts,  ACH  and  ACD.    The  part 

IaCB   bearing   upon   the   part   ACD 

Ipresses  it  witli  its  whole  weight;  nor   B| 
can  the  part  ACD  sustain  this  press- 
ure, and  continue  unmoved,  if  it  is 

{not  opposed   by   an    equal   contrary 

I  pressure.     And  therefore   the  parts 

|e({ually   press    each    other   by   their 
weights — that  is,  equally  attract  each 

other,  according  to  the  third  law  of  motion;  and,  if  separated 
and  let  go,  would  fall  towards  each  other  with  velocities  re- 
ciprocally as  the  bodies.  All  which  we  may  try  and  see  in  the 
loadstone,  whose  attracted  part  does  not  propel  the  part  at- 
tracting, but  is  only  stopped  and  sustained  thereby. 

Suppose  now  that  ACB  represents  some  small  body  on  the 
earth's  surface ;  then,  because  the  mutual  attractions  of  this 
particle,  and  of  the  remaining  part  ACD  of  the  earth  towards 
each  other,  are  equal,  but  the  attraction  of  the  particle  towards 
the  earth  (or  its  weight)  is  as  the  matter  of  the  particle  (as  we 
have  proved  by  the  experiment  of  the  pendulums),  the  at- 
traction of  the  earth  towards  the  particle  will  likewise  be  as 
the  matter  of  the  particle;  and  therefore  the  attractive  forces  of 
all  terrestrial  bodies  will  be  as  their  several  quantities  of  matter. 
The  forces  (prop.  71,  book  I),  which  are  as  the  matter  in 


Fig.  b 


*  [Maskelyne  {^\)mys  with  reference  to  this :  "It  will  appear,  Iry  a  very  easy 
calculation,  i/iat  such  a  mountain  would  attract  Vie  plumb-line  1'  IH"  from  the 
perpendicular."] 

f  [  TViis  jxiragraph  is  of  great  imjyortance,  becavse  in  it  Newton  indicates 
the  metliods  of  all  tlie  experiments  yet  made  in  order  to  measure  gravitational 
attraction  in  terrestrial  bodies.] 

B  17 


MEMOIRS    ON 


I  i.ii 


n: 


'  i. 


i  I 


f-l 


m 


r.i 


terrestrial  bodies  of  all  forms,  and  therefore  are  not  mutable 
witii  tiie  forms,  must  be  found  in  all  sorts  of  bodies  whatsoever, 
celestial  as  well  as  terrestrial,  and  be  in  all  proportional  to  their 
quantities  of  matter,  because  among  all  there  is  no  dilteretice 
of  substance,  but  of  modes  and  forms  only.  But  in  celestial 
bodies  the  same  thi?ig  is  likewise  proved  thus.  We  have  shewn 
that  the  action  of  the  circumsolar  force  upon  all  the  planets 
(reduced  to  equal  distances)  is  as  the  matter  of  the  planets; 
that  the  action  of  the  circumjovial  force  upon  the  satellites  of 
Jupiter  observes  the  same  law;  and  the  same  thing  is  to  be  said 
of  all  the  planets  towards  every  planet;  but  thence  it  follows 
(by  prop.  09,  book  I)  that  their  attractive  forces  are  as  their 
several  quantities  of  matter. 

As  the  parts  of  the  earth  mutually  attract  one  another,  so 
do  those  of  all  the  planets.  If  Jupiter  and  its  satellites  were 
brought  together,  and  formed  into  one  globe,  without  doubt 
they  would  continue  mutually  to  attract  one  another  as  before. 
And,  on  the  other  hand,  if  the  body  of  Jupiter  was  broken  into 
more  globes,  to  be  sure,  these  would  no  less  attract  one  another 
than  they  do  the  satellites  liow.  From  these  attractions  it  is 
that  the  bodies  of  the  earth  and  all  the  planets  effect  a  spheri- 
cal figure,  and  their  parts  cohere,  and  are  not  dispersed  through 
the  aether.  But  we  have  before  proved  that  these  forces  arise 
from  the  universal  nature  of  matter  (prop.  72,  book  I),  and 
that,  therefore,  the  force  of  any  whole  globe  is  made  up  of  the 
several  forces  of  all  its  parts.  And  from  thence  it  follows  (by 
cor.  3,  prop.  74)  that  the  force  of  every  particle  decreases  in 
the  duplicate  proportion  of  the  distance  from  that  particle; 
and  (by  prop.  73  and  75,  book  I)  that  the  force  of  an  entire 
globe,  reckoning  from  the  surface  outwards,  decreases  in  the 
duplicate,  but,  reckoning  inwards,  in  the  simple  proportion  of 
the  distances  from  the  centres,  if  the  matter  of  the  globe  be 
uniform.  And  though  the  matter  of  the  globe,  reckoning  from 
the  centre  towards  the  surface,  is  not  uniform  (prop.  73,  book 
I),  yet  the  decrease  in  the  duplicate  proportion  of  the  distance 
outwards  would  (by  prop.  76,  book  I)  take  place,  provided  that 
difformity  is  similar  in  places  round  about  at  equal  distances 
from  the  centre.  And  two  such  globes  wiP  (by  the  same  prop- 
osition) attract  one  the  other  with  a  force  decreasing  in  the 
duplicate  proportion  of  the  distar'^e  between  their  centres. 

Wherefore  the  absolute  force  of  every  globe  is  as  the  qnan- 

18 


TIIK    LAWS    OF    GRAVITATION 


0  not.  mutable 
ios  whatsoever, 
rtional  to  their 

1  no  diirerence 
ut  in  celestial 
V^e  have  shewn 
ill  the  phinets 
I'  the  jihmets; 
he  satellites  of 
ig  is  to  be  said 
nee  it  follows 

i  are  as  their 

le  another,  so 
satellites  were 
without  doubt 
her  as  before. 
8  broken  into 
tone  another 
raotions  it  is 
feet  a  spheri- 
rsed  through 
J  forces  arise 
book  I),  and 
de  up  of  the 

follows  (by 
decreases  in 
lat  particle; 
of  an  entire 
eases  in  the 
roportion  of 
he  globe  be 
zoning  from 
V-  73,  book 
he  distance 
ovided  that 
1  distances 
same  prop- 
ping in  the 
ientres. 

the  quan- 


lityof  matter  which  the  globe  contains;  but  the  motive  force 
hv  which  every  globe  is  attracted  towards  another,  and  which, 
in  terrestrial  bodies,  we  commonly  call  their  weight,  is  as  the 
coiitont  under  the  quantities  of  matter  in  both  globes  applied 
to  the  square  of  the  distance  between  their  centres  (by  cor.  4, 
prop.  ?(>,  book  I),  .J  which  force  the  quantity  of  motion,  by 
which  each  globe  in  a  give!i  time  will  be  carried  towards  the 
other,  is  proportional.  And  the  accelerative  force,  by  which 
every  glol)e  according  to  its  quantity  of  matter  is  attracted 
towards  another,  is  as  the  quantity  of  matter  in  that  other  globe 
applied  to  the  square  of  the  distance  between  the  centres  of 
the  twi  (by  cor.  2,  prop.  76,  book  I),  to  which  force  the  ve- 
locity by  which  the  attracted  globe  will,  in  a  given  time,  be 
carried  towards  the  other  is  proportional.  And  from  these 
principles  well  understood,  it  will  be  now  easy  to  determine 
the  motions  of  the  celestial  bodies  among  themselves. 


Sir  Isaac  Newton  was  born  at  Woolsthorpe,  near  Grant- 
ham, in  Lincolnshire,  in  1645J.  He  was  educated  at  the  Grant- 
ham grammar-school,  entered  Trinjty  College,  Cambridge,  in 
1661,  and  received  his  degree  four  years  later.  He  at  once 
began  to  make  those  magnificent  discoveries  in  mathematics 
and  physics  which  have  made  his  name  immortal.  In  1665  he 
committed  to  writing  his  first  discovery  on  fluxions,  and  shortly 
afterward  made  the  unsuccessful  attempt,  to  which  we  have 
already  referred,  to  explain  lunar  and  planetary  motions.  He 
next  turned  his  attention  to  the  subject  of  optics  ;  his  work  in 
that  field  includes  the  discovery  of  the  unequal  refrangibility 
of  differently  coloured  lights,  the  compositeness  of  white  light 
and  chromatic  aberration.  Having  erroneously  concluded  that 
this  aberration  could  not  be  rectified  by  a  combination  of  lenses, 
he  turned  his  attention  to  reflectors  for  telescopes  and  made  a 
great  advance  in  that  direction.  His  name  is  also  closely  iden- 
tified with  the  colours  due  to  thin  plates.  From  1669  to  1701 
he  was  Lucasian  professor  of  mathematics  at  Cambridge.  He 
was  elected  to  membership  in  the  Royal  Society  in  1671,  and 
from  1703  until  his  death  was  its  president ;  he  became  a  mem- 
ber of  the  Paris  Academy  in  1609.  The  publication  of  his  work 
on  Optics  had  caused  some  controversy,  and  such  a  lover  of 
peace  was  Newton,  and  so  little  did  he  care  for  the  praise  of 

19 


^rmmmm!^^ 


ME  MO  I  Its    ON    THE     LAWS    OK    (i  It  A  V  IT  A  T  I  O  N 


' 


i 


'■'■\ 


:        ,,t 


p!!l 
Mi:; 

•ij'i 


:  r 


I  t  ! 


!  it 


the  world,  tluit  it  was  only  iit  tlio  onniost  solicututioii  of  Ilalley 
tiiat  ho  was  williiii^  to  give  to  tlio  public  tlio  rosults  of  liis  won- 
derful resoarohes  on  coiitral  orbits,  and  universal  ^gravitation  ; 
these  included  an  explanation  of  the  lunar  inequalities,  the 
figure  of  the  earth,  the  precession  of  the  equinoxiis  and  the 
tides,  and  a  method  of  comparing  the  masses  of  the  heavenly 
bodies.  In  lOOl)  he  became  a  member  of  Parliament,  in  109(5 
Warden  of  the  Mint,  and  from  lOU'J  until  his  death  was  Master 
of  the  Mint,  lie  gave  much  valuable  aid  in  the  rccoimige  of 
the  money  and  in  questions  of  finance  at  this  period,  lie  was 
knighted  in  1705.  During  the  latter  years  of  his  life  much  of 
his  time  was  devoted  to  his  public  duties.  Ho  died  in  1737, 
and  was  buried  in  Westminster  Al>bey. 


If 


LA  FIGURE  DE  LA  TERRE 

netermint'e  par  les  Obscrvivtioiis  do  Messieurs  Boiiguer,  et 
(le  la  Coudarniiio,  do  rAeadeinie  Uoyalc  des  S<;ieiHU's,  eiivoyes 
|)iir  ordre  du  Roy  au  Peroii,  pour  observer  aux  environs  do 
l'f^(juatenr. 

Avec  uno   Uelatiou  abregee  de  ee  Voyage,  qui    eontiont  la 
description  du  l*ays  dans  lequel  les  optM-ations  ont  etc  faites. 

Pau  M.  BOUGUER 
X    Paris,  1749 


m 

MM 


Secliou  7.  pp.  327-394 


THE  FIGURE  OF   THE  EARTH 

Determined  by  the  observations  of  MM.  Bouguer  and  de  la 
Condaniine,  of  the  Royal  Academy  of  Sciences,  sent  to  Peru 
by  order  of  the  King  to  make  observations  near  the  equator. 

With  a  brief  account  of  their  travels  and  a  description  of  the 
country  in  which  the  investigations  were  made. 

By  PIERRE    BOUGUER 

Paris.   1749 


id 


w 


fyl5 


Section  7    pp.  327-394 


21 


3' 


1 

I 


k  i 


iif   ! 

,1     ' 


>'  itfl! 


Ji 


CONTENTS  OF  SECTION  VII 

PAOR 

Jntroditction 23 

Chap.  I, — Exjteriments  Made  in  Order  to  find  the  Length  of  the  Seconds- 
Pendulum  24 

Description  of  Pendulum 24 

Method  of  Observation  ( Omitted). 

Observed  Ijenyths  of  Seconds- Pendulum  at  Various  Places 25 

Corrections  to  be  Made  in  the  Observed  Lengths 25 

Corrected  Lengths  <f  Seconds-Pendulum  at  Various  Places. ...  27 
Chap.  11. — Comjuirison  of  Attraction  and  Centrifugal  Force  (Omitted). 
Chap.  III. — liemarks  on  the  Diminution   in  Attraction  at  Different 

Heights  above  Sea-level 27 

Calculation  of  the  Attraction  Due  to  a  Plateau 29 

Deduction  of  the  Mean  Density  of  the  Earth  from  Pend- 
ulum E.vperiments 32 

Chap.  IV. — On  the  Deflection  of  the  Plumb  line  by  a  Mountain 33 

Description  of  Mount  Chimborazo 34 

Its  Deflection  of  the  Plumb-line  Calculated  from  the  Theory. .  34 

Various  Ways  Suggested  for  Showing  the  Deflection 35 

Description  of  the  Method  Employed 38 

Examination  of  tfie  Attraction  of  Chimborazo 39 

Meridian  Altitudes  at  the  First  Station 40 

Measurements  Made  toflnd  the  Itelative  Positions  of  the  two 

Stations 41 

Meridian  Altitudes  at  the  Second  Station  {Omitted). 

Collected  Meridian  Altitudes  at  the  Second  Station 42 

Calculations  for  the  Observed  Deflection  of  the  Plumb-line. . .  42 

Its  Poor  Agreement  with  that  Calculated  from  Theory 43 

Appendix  {Omitted). 


22 


iiii 


PAOK 

.  .  •  • 

23 

)ndH- 

•  •  •  • 

24 

.... 

24 

25 

35 

s.  .  .  . 

27 

ted). 

f'erent 

27 

29 

Pend- 

32 

.   ... 

33 

34 

ovy . . 

34 

35 

88 

39 

40 

e  iwo 

41 

42 

ne. . 

42 

.  43 

SECTION    VII.    OF    B0UGUER^8    FIGURE 

OF    THE    EARTH 

ACCOUNT    OF    THE    KX  PKIU  MENTS   OR    OBSERVATIONS    ON    GRAV- 
ITATION,   WITH    REMARKS    ON    THE    CACSES    OF 
THE    FIGURE   OF   THE    EARTH 

1.  Having  discussed  evorytliitig  that  bears  on  the  earth  con- 
sidered as  a  geometrical  body,  it  remains  for  us,  before  terminat- 
ing this  wori<,  to  verify  the  facts  whicli  give  us  some  slight 
knowledge  of  the  interior  conformation  of  this  great  mass  con- 
sidered as  a  physical  body.  .  .  . 

2.  The  first  question  which  presents  itself  on  this  matter  is 
ji  consideration  of  the  part  played  in  the  flattening  of  tlie  earth 
by  the  attraction  which  compresses  it  from  all  sides,  urging  all 
masses  towards  certain  points.  We  know,  since  M.  Richer  first 
remarked  it  (in  Vill'Z  in  Cayenne),  that  this  force  is  not  every- 
where the  same.  It  is  greater  towards  the  poles,  and  less  to- 
wards the  equator.  This  agrees  perfectly  with  the  figure  of  the 
earth,  which  appears  to  have  yielded  a  little  to  the  great  press- 
ure at  the  poles,  and  to  be  slightly  elevated,  on  the  contrary, 
at  the  equator,  where  the  compressing  force  was  more  feeble. 
But  does  the  eifect  correspond  exactly  to  the  cause  upon  which 
we  desire  it  to  depend?  Is  the  difference  in  attraction  so  great 
that  we  can  attribute  to  it  all  the  inequality  which  exists,  as 
we  have  seen,  between  the  two  diameters  of  our  globe  ?  To 
answer  this  question  it  is  necessary  to  determine,  by  exact  ex- 
periment, how  much  the  attraction  actually  differs  in  different 
})arts  of  the  earth.  .  .  .  We  have  two  methods  for  observing 
tlie  change  in  attraction  as  we  pass  from  one  region  to  another  ; 
we  have  only  to  examine  how  much  more  quickly  or  more 
slowly  a  pendulum  of  given  length  oscillates ;  or  else  to  find 
the  length  of  the  pendulum  whose  time  of  vibration  is  exactly 

23 


MKMolliS    ON 

II  spcoiul  ;  the  (lifTorcnros  whic^h  wo  shall  tiiid  in  the?  h'fi^th  nf 
this  pciiiiiilniri  will  (Ictcrfniiio  the  chuii^eH  uf  the  uttructioii  us 
we  go  from  one  region  to  another. 


I 


Ii! 
'  i 

1 

1                     ? 

1'    i 
i  1 

1  ' 

1          1 

N' 

j  ' 

!  i 

! 

; 

1 
i 

1 

i 

1 

1 

i'      1 

\         i 

i 

i;         i 

f-       1 
f       •' 

j 

1 

'    ''       i 

t    ' 

■ 

t: 

1 

ti 

ACCOl'NT    OK    THK    KXI'KIMMKNTS     MADK    FOR   THK    PUU|M)SK    OF 
I>KTKKMININ(}    TIIK    I.KNOTM    OF   THK    HK(;ON  DS-I'FN  r)r  LTM 

.'J.  My  lirst  experiments  witli  tiie  pendulum  were  niiulo  iit 
I*etit-(ioiivo  in  the  ishmd  of  St.  Dominguc!.  'I'iiey  iire  reported 
in  the  memoirs  of  the  Academy  for  1TIJ5  and  17JJ<».  .  .  . 

4.  'ri)e  instrument  whieli  I  almost  always  used,  and  which  I 
still  use,  is  extremely  i-imple.  I  make  the  pendulum  always 
exactly  of  the  same  len^t!»..  and  I  compare  its  osirillations  with 
those  of  a  clock  which  I  regulaie  hy  (laily  observations.  Jt  is 
not,  properly  speaking,  by  the  diiferent  lengths  of  the  ])end- 
nlum  that  1  judge  of  the  intensity  of  gravitation  at  different 
places  ;  I  judge  of  it  only  by  the  greater  or  less  nipidity  of  the 
oscillations,  or  by  the  number  of  oscillations  made  by  the  pend- 
ulum in  24  hours.  ...  It  appears  to  me  to  be  much  easier  to 
count  the  number  of  oscillations  than  to  measure  directly  dif- 
ferences of  a  few  hundredths  of  u  line*  in  the  length  of  the 
pendulum. 

[Then  follows  an  account  of  his  pciKhdum.  The  boh  was  of 
copper,  composed  of  two  equal  truncated  cones  joined  at  their 
greater  bases.  The  thread  was  a  fibre  of  aloe,  which  is  not  af- 
fected by  the  weather.  The  length  was  maintained  constant  by 
having  it  always  so  that  an  iron  ride  just  fitted  in  between  the 
clamp  and  the  bob.  The  length  of  the  equivalent  simple  pendulum 
was  3(1  pouces,  7.015  lines. 

Botiguer  gives  a  description  of  a  scale  fixed  behind  the  pend- 
ulum, by  means  of  which  he  could  observe  the  decrement  and  the 
time  required  by  the  pendulum  to  gain  an  oscillation  on  the 
clock.  ] 

10.  It  is  time  to  relate  the  experiments.  ...  I  shall  choose 
one  of  those  which  I  made  on  the  rocky  summit  of  Pichincha 
[2434  toises  above  sea-level],  in  the  month  of  August,  1737.  The 

*  [73  pouces  =  1  torn  =  1.949  metres  =  6.3945  ft.    13  lines  =  1  pouce  ] 

34 


Til  K    LAWS    (H-    (',\l\\  I  r.\  rioN 


t'orco  of  attnictioii  was  fi'd))*?,  not  only  btM-ausc  we  wcrr  lu-aily 
ovor  tlio  r(|iiatoi'  at  tliis  place,  liiil  also  ln'caiisc  wo  were  ut  u 
very  ;;;n'al.  Ii«'i.nlit  ahove  the  siirlace  ul'  the  8eu.  .  .  . 

I  Dihtils  of  c.r/irriiHrnt.  | 

I",'.  .  .  .  We  liinl  in  thin  way  that  the  pendulum  wliieli  heat.s 
secomls  at  tiie  equator,  and  in  the  hi^^Miest  aecu'SsihU^  plaei;  on 
the  eartli,  is  IJll  polices  (LOW  lines  in  leiij^th.  I  inatle  other  ex- 
periments at  the  same  place  which  a;,'r»'e'i  as  exactly  as  possihie 
with  this  result.  \()ih'  mmlr  hij  Ihni  Aiitoxiitdv  t'l/an  i/air  'M\ 
pnnrvs,  (I.T15  lines.  \Vc  may  lake  us  llie  mean  'M\  jtouci's,  O.io 
lint's.  I 

i;{.  I  have  found  by  the  same  proeeedini^s  ami  with  the  aid 
of  the  same  instriiimints,  the  len;^4h  of  the  s(H3onds-pendiiliim 
at  t^iiito  I  14<l(l  laisrs  ul/oiw  sm-lcirl],  to  be  'M\  ponces,  (t.H'i  or 
(l.s;{  lines.  1  have  verified  it  at  ditterent  times  and  in  all  sea- 
s«)ns  of  the  year:  at  times  of  aphelion  and  perihelion,  at  the 
e(|iiino\es,  and  when  the  sun  was  at  intermediati^  points;  the 
extreme  results  were  'M\  pouees,  O.T'.J  lines  and  ().8r>  lines,  with 
no  dilTerciKies  whioh  could  not  bo  attributed  to  the  inevitable 
errors  of  observation.  .  .  . 

I  '/'//<'  (jHCstion  of  n  possible  f/cnrlt/  cJuDtf/e  is  disrnssed. 

HjjU'vinwnts  were  nuale  irilh  the  same  ((jtjxtrahis,  in  1T4(>,  (tt 
I' hie  tie  r /nra.  14'  or  i^i'  froni  the  et/aafor,  ((nd  searcehi  40  toises 
above  sea-lerel.  /ioaf/aer  regards  this  determination  as  that  of 
the  true  e(/ainoctial pendalnm.] 

ir). 


Place 

I.OIl),'!!! 

foiiiMl  liy  oxporlmonl 

I  2484  toises  absolute  lieiglit. 
Under  the  equator  ul  •]  1460     " 

/  Sen-level 

36  pfHices, 

6.70  liiH's. 

683      " 

7  07      " 

At  Poitnliell  .,  9"  34'  N.  latitude 

7.16      •• 

At  l»eiii-0(»iive,  18°37'  "         " 

*  * 

7.33      " 

At.  Puiis       

8.58      " 

CORRECTIONS    WHICH     MUST     BE    API'LIKF)    TO    TlIK     LHNGTIl    OF 

TIIE    PENDULUM    AS    DETKUMIXKI)    DIRECTLY    FROM 

THE    EXI'KRIMKNTS. 

16.   [novf/uer  remarks  that  these  eorrertions  arise  from  chanc/cs 
in  tenwerature  and  in  the  const  it  at  ion  of  t  lie  atmosphere.]     The 

85 


1 

i 

1 

,1 

■ 
t 

1 

■  ! 
'  1 

1 

,         1 

',   t 

j 
i 

r| 

1 

IH.H 


fini 


I'!!'! 


M  KMOI KS    ON 

firHt  niiiHc  (loos  not  roftlly  rlmtipc  tlio  length,  it  only  iniikoH  it 
uppciir  <litT<'r<Mit  lUH^onliii/^  Jis  tlic  niciisureH  wo  iiho  un*  <lilTor- 
cntly  iilt<u'«M|  hy  licjit  or  ('(jid;  hiit  the  otiior  ciiusf  brink's  in  ii 
roul  iniM|iiiility,  simu!  it  prodiiccK  nmrly  the  Huinu  etToct  uh  if 
tho  weight  wvrv.  ^rnitcr  or  HtimlliT.  .  .  . 

IT.  .  .  .  Sinco  tlu)  t('in|K'riitnr(' of  Quito  (loos  not  (litTor  from 
thiit  of  {'aris  in  the  rnitldio  (tf  sprin;,',  wo  huvo  only  to  refer  all 
our  results  to  it.  'i'hiit  is,  without  altori.ig  the  lengths  of  the 
pendulum  found  in  these  two  cities,  we  huvo  only  to  (M)rre(!t  nil 
the  others  hy  in(!reasin<i  or  diminishing  them,  aoeording  as  the 
metal  rulos  we  used  wore  e,xpund(M|  hy  the  hea^  or  contraotod 
hy  the  cohl.  (  /A'  ronrluilvs  finni  his  r.rperimcnfs  llutt  a  r/ianf/(', 
of  tenffffi  of  pcH(/it/inu  of  .0^  litirs  rorrcspiunis  fo  <t  <'linn(jp  of 
ffimpcrahnr  of  'X'  li.  Uvncv  hi  had  to  mid  .OT.")  linos  lo  I  ho  lfinf/lh 
found  at  sfff-lcrcl,  (Hitl  stihh'orl  .0")  lines  from  Ihal  found  at 
/'irhinrhd.  | 

18.  Thoro  is  little  more  diflieulty  in  finding?  the  altoratioii  in 
the  hfUfifth  of  the  p(M'.duluni  eaused  by  the  nuMlium  in  which 
the  experiments  are  made.  This  medium, whether  rare  or  dense, 
has  a  certain  weight,  atid  that  of  the  small  nuiss  of  copper,  of 
which  the  hob  of  tho  pendulum  is  formed,  is  a  little  lessened 
by  it.  The  small  mass  tends  to  fall  to  the  earth  with  only  the 
excess  of  its  weight  above  that  ot  the  air  which  surroutuls  it. 
MMius  our  pendulums  are  acted  on  by  a  forte  a  little  less  than  if 
we  had  jierformcd  tho  oxperitnents  in  vnruo :  and  the  lenjijth 
of  the  seconds-pendulum,  wliicdi  wo  found  directly  from  experi- 
ment, is  a  little  too  short  in  tlie  same  proportion. 

10.  The  use  of  the  barometer  emibles  us  to  find  the  ratio  be- 
tweeti  the  weight  of  mere. try  and  of  air  in  all  tho  parts  of  the 
atmosphere  which  are  accessible.  We  observe  how  many  feet 
it  is  necessary  to  ascend  or  descend  in  order  to  change  the 
height  of  the  mercury  by  a  lino.  ...  I  have  found  in  this  way 
that  it  was  only  necessary  to  express  the  first  (tho  weight  of 
air)  by  unity,  at  (h  j  summit  of  Pichincha,  if  one  expressed  that 
of  copper  by  IK'OO.  ...  So  I  always  found  the  seconds-pend- 
ulum too  small  by  jT^^ivT^th  part.  To  correct  for  this  error 
we  must  add  .04  lines  [at  Pichincha ;  .05  at  Quito;  .06  at  sea- 
level.^  .  .  .  This  is  the  first  time  that  any  one  has  taken  ac- 
count of  this  small  correction  which  enters  into  tho  experi- 
ments, but  we  catniot  neglect  it  if  wo  wish  to  attain  tho  greatest 

accuracy.  ... 

26 


Til  i:    LAWS    (H'    <i  liA  Vn  ATluN 


I  /linii/Hcr  f/ifft  proves  Hial  Ihv  limv  of  rihnilinn  is  unf  n/tpn'ri- 
tihljl  nlf'nfctf  hif  till'  nsishnirv  of  thr  tiir.  |* 

•,'•.'.   ('orrcftcMJ  l»'ii;,'tlis  nf  jlic  s( nds-jR-iitliiliim,  or  siicli  \\n 

tlicy  would  l)u  if  tlio  oHcillutions  wwv  iiiailc  ///  rtniio. 


I'liiro 


t  2484  tolfli'H  iili!4M|iiie  height. 
I  ikIjt  tli«  i-qimtor  Hi  \  146(1 

f  Sni-lcv«'l        


At  Portohcllo.  9'  W  N.  lalltiid.; 
A I  IVtil  O..UVI'.  18^27  "  "  . 
At  PariM 


:)6  pollers.  0.01)  lines 
7'Jl     " 

7.;u)    " 

"  '*        7  47     ' ' 

"         "        H.07     " 


i 


II 

COMPAHISON     or    ATTRACTION     AND    TIIK    CKNTUIFlf}  A  I,    TOKCK 

WIIK  11    H>I)!KS    ACt/riKK   l«Y   TIIK    MOTION    O I'  TIIK    KAKTII 

A  HO  IT    ITS    AXIS,   WITH     K  KM  AUKS    ON    TIIK    KFI'KCTS 

Ol"    THKSK    TWO    I'OKCKS. 

I  lioHi/urr  ^'ti«is  lh(tt  I  he  prinn'/in-  (iffntcfiott  {fluff  (s,  f/tr  affnir- 
fioN  flic  mrfli  irotifd  Ihnw  if  if  tri'ir  af  rrsf)  is  fit  fhv  cvufvifuijai 
force  (IS  •/iSSJil  :  1.  /fc  i/iirs  n  fitfjfr  slioirimj  flw  ifirrcasr  in  flie 
tvnijfli  of  flw  siroiiifs-pfHilnhnti  iif  various  lafifuilcs,  due  fo  flic 
vcnfrifiKjnl  forvp.  The  folio iriuij  hvadiiKis  will  (fiiw  an  idea  of 
llie  Hiaffcr  ronfirned  in  flie  resf  of  fliis  rlmpfcr.  \ 

The  centrifugal  force  produced  by  the  riiotioti  of  tlie  earth 
about  its  axis  is  not  sufficient  to  produce  the  observed  differ- 
ences in  weight. 

The  primitive  attraction  does  not  tend  towards  a  common 
point  as  centre. 


t  >a 


III 

REMARKS   ON    THE    DIMINUTION    IN    THK    ATTRACTION    AT    DIF- 
FKRENT    HEIGHTS    ABOVE    THE    LEVEL   OF    THK    SEA. 

40.   The   experiments  witli   the    pendulum   which   we   have 
made  at  Quito  and  on  the  summit  of  Fichincha  teach  us  that 

*  [ike  note  at)  puije    06] 
27 


MKMOlllS    ON 


Hi 


tlio  iittnictioii  oliaiiiifos  with  tlio  distimco  from  tlio  centre  of  the 
ojirtli.  'riiis  force  goes  on  tlirniiiishiiig  sih  we  Jisceiitl ;  I  have 
found  thi!  pendulum  iif  Quit(»  to  he  shorter  than  at  sea-level  by 
.'V'l  lines,  or  the  iVsyth  part:  and  in  mount'iig  to  the  summit  of 
Pichincha  the  pendulum  is  shortened  again  ijy  ,11)  lines,  and  is 
(^lsth  part  shorter  than  at  sea-level.*  One  cannot  attribute 
these  differences  to  the  centrifugal  force,  which,  being  greater 
the  higher  we  ascend,  ought  to  diminish  a  little  further  the 
primitive  attraction.  'IMie  centrifugal  force  is  increased  by  the 
lieight  of  the  mountain  l)y  the  T-jVit*'''  P''-''^  <^>'»ly>  smd  J^s  it  is 
itself  but  the  jjj>th  part  of,  the  weight,  it  is  clear  that  its  new 
incroasi!  (!orresponds  to  .001  lines  only  in  the  length  of  the 
pendulum,  and  oo  does  not  sensibly  coi^tribute  to  the  dimi- 
nution of  the  other  force. 

41.  If  we  compare  the  shortening  which  the  pendulum  re- 
ceives with  the  heigiit  at  whi(di  the  experiment  was  made,  we 
see  that  the  forces  do  not  decrease  in  the  simple  inverse  ratio 
of  the  distances  from  the  centre  of  the  earth,  but  that  they 
follow  rather  the  proportion  of  the  square.  Quito  is  14()(J 
toises  above  sea-level,  or  ^^Vt^^'  "'  ^^^*^  radius  of  the  eartli  ;  but 
it  has  been  found  that  the  attraction  is  less  by  a  fraction  much 
more  considerable — mimely,  by  a  itj^t^I^  part,  which  is  nearly 
double  :  this  is  not  verv  far  from  the  inverse  ratio  of  the 
square  of  the  distance.  .  .  .  We  have  a  seconi'.  example  in  the 
experiment  made  on  Pichincha.  The  absolute  height  of  this 
mountain,  which  is  2434  toises  above  sea-level,  is  j-jVs^'^  ^^  the 
radius  of  the  earth.  The  diminution  of  the  length  of  the  pend- 
ulum, or  of  the  attraction,  ought  then  to  be  the  jj^^jth  part, 
if  it  is  to  be  in  the  inverse  ratio  of  the  square  of  the  distance  ; 
but  it  was  by  no  means  so  great — in  fact,  only  the  g^th  part. 

42.  This  diminution  in  attraction,  as  we  go  above  sea-level,  is 
quite  in  conformity  with  what  we  otherwise  know.  We  can 
compare  with  the  attraction  here  experimented  upon  that 
which  keeps  the  moon  in  its  orbit,  or  wnich  obliges  it  con- 
tinually to  perform  a  circle  about  us.  These  two  forces  are 
exactly  in  the  inverse  ratio  of  the  squares  of  the  distances 
from  the  centre  of    the  earth.     We  can   make  the  same  ex- 

*  [Pendulum  obsterratiom  were  nvule  ni  these  and  other  places  in  Peru  hy  df 
la  Condnmine  also  (H,  pp.  70,  144,  162-1G9).  For  a.  complete  hihliography  of 
pendnlitm  e.rperiments,  see  that  published  by  La  Societe  Franfaise  de  Physique 
(178,  ml.  4)  1 

28 


TIIK    LAWS    OI'    <;UAVITATI(>N 


iiiiiiiiJitioii  with  rospoft  to  tlio  principal  phiiiots  wiiicii  liavo 
suvt'i'iil  satolliti's,  or  with  respect  to  the  siiii,  to\var<ls  wliich 
ill!  tlie  })rincipal  phiiiets  are  attracted,  ami  we  shall  always 
liiid  the  law  of  the  square.  Why,  then,  do  our  experiments 
constantly  give  a  law  not  entirely  in  agreement  with  this? 
Is  it  necessary  to  attrihnte  the  diirerence  to  some  error  on 
our  j)art  ;  or  can  it  be  that  in  the  m'ighborhood  of  great  masses 
like  the  earth  the  law  under  consideration  is  oi)served  in  an 
imperfect  manner  only  ? 

4IJ.  We  shall  find  ourselves  in  x  position  to  solve  this  dilli- 
culty,  perhaps,  by  remarking  that  the  Cordilleras,  on  which  we 
were  placed,  form  a  kind  of  [)lateau,  or,  what  in  certain  ways 
amounts  to  the  sajne  thing,  the  surface  of  the  earth  is  there 
carried  to  ii  greater  height  or  to  a  greater  distance  from  the 
centre.  Tiiere  is  reason  for  believing  that  in  this  second 
case  the  attraction  would  be  a  little  greater  ;  for  it  is  natural 
to  think  that  it  depends  upon  the  size  of  the  attracting  mass. 
There  are  then  two  things  to  be  considered  in  the  cas((  of 
the  experiments  on  the  pendulum  which  1  have  reported. 
'IMiese  experiments  were  made 
at  a  great  height  above  the  av- 
erage surface  of  the  earth,  and 
therefore  the  attraction  ought 
to  be  found  a  little  less.  But, 
on  the  other  hand,  the  group  / 
of  mountains  on  which  Quito  is    j 


placed  and  on  which  Pichincha 
rises,  and  ail  the  other  sum- 
mits to  which  it  acts  as  a 
plinth,  ouglit  to  produce  nearly 
the  same  elfect  as  if  the  earth 
at  this  place  were  larger  or  had 
a  greater  radius.  The  attrac- 
tion on  this  account  ought  to  increase.  Thus  it  depends  on 
a  kiml  of  chance,  or,  to  speak  niore  ])hiloso])hically,  it  de- 
pends on  circumstances  which  we  do  not  yet  know,  whether 
the  attraction  at  Quito  will  be  equal  to  that  at  sea-level,  or  be 
smaller  or  larger. 

44.  Suppose  that  the  circle  ADD  represents  tlie  circum- 
ference of  the  earth,  of  which  C  is  the  centre,  and  that  Ar^ 
is   the   amount    by  which   Quito,  situated    at    a,   is    elevated 

89 


;i  - 


M  E  M  O  I  R  S    O  X 


i; 


above  spa -level.  Inia^nne  a  now  splierieal  >«bell  of  terrestrial 
matter,  occiipyin^  nil  the  interval  between  the  two  concen- 
tric surfaces  ADD  and  ridd ;  or,  which  conies  to  the  same 
thing,  imagine  that  the  eartii  increases  in  radius,  and  that 
Quito,  without  changing  its  position,  remains  at  the  level 
of  the  sea,  now  supposed  much  higher.  There  is  every  reason 
to  think  that  the  attraction  at  Quito  would,  as  a  consequence, 
be  found  greater  than  it  actually  is  at  A  or  at  D,  in  the 
ratio  of  CA  to  Ca.  It  is  necessary  for  that,  however,  to  sup- 
pose that  the  layer  of  earth  enclosed  between  the  two  con- 
centric surfaces  is  of  the  same  density  as  all  the  rest ;  for  if 
the  density  were  different  the  increase  would  no  longer  be  in 
the  same  ratio. 

45.  Call  /•  the  radius,  and  A  the  density  of  the  earth.  Then 
rA  is  the  attraction  at  all  the  points  A,  D,  etc.,  suppo^-^ing 
that  the  earth  ends  there.  Call  h  the  height  Aa,  which  is 
very  small  compared  with  r.  Then  the  attraction  at  a  is 
less  than  at  A,  in  the  ratio  of  r'  :  (r  +  //)',  or  its  diminution 
will  be  as  2h  :r;  that  is,  if  the  attraction  is  rA  at  A,  it  is 
(r  — 27a)A  at  a,  and  this  supposes  that  the  earth  has  CA  only 
for  effective  radius.  But  all  this  will  be  subj>'C!  \  ?  change 
if  we  add  to  our  globe  the  layer  Ar/D,  whose  density  is  h. 
This  new  spherical  laye/,  if  it  had  the  same  density  as  the 
rest,  would  augment  the  attraction  at  the  surface  in  the  same 
ratio  as  the  radius  of  the  earth  became  greater.  The  increase 
would  be  in  the  ratio  of  r  :  r  -^h. 

46.  Thus  the  added  layer  would  not  only  make  up  for  the 
decrease  which  the  attraction  actually  suffers  when  we  go 
away  from  the  earth,  in  rising  by  the  height  Ka  —  hy  but  would 
add  a  new  amount  to  it,  equal  to  half  the  diminution,  since 
it  would  make  this  attraction,  which  is  actually  r  —  2h  at  the 
point  a,  become  r-{-h.  It  follows  that  the  attraction  which 
the  spherical  layer  can  produce  at  its  exterior  surface  ai  i-^ 
is  expressed  by  dh,  or  three  times  its  thickness;  but  \V'3  mv.A 
multiply  by  the  density  S,  because  we  suppose  that  the  den 
sity  of  the  layer  and  that  of  the  earth  as  a  wholo  are  not 
equal. 

47.  To  recapitulate:  When  the  earth  has  its  radius,  CA  =  r, 
the  attraction  at  A  is  ?-A,  and  at  the  height  h  is  {r  —  2h)A. 
But  when  we  add  to  the  earth  the  spherical  layer  AdD,  the 
attraction  at  a  becomes  (r  — 2/i)A+3//3. 

30 


THE  LA  w s  ( ) K  ( ;  II A \  1  r a  r i o n 

48.  All  that  remains  now  to  he  remarked  is  that  the  Cordill- 
eras of  Peru,  however  j^reat  they  may  he,  ought  not  to  produce 
tlie  same  effect  as  the  spherical  shell  which  we  have  assumed. 
If  the  base  EE  of  the  Cordilleras  were  exactly  double  its 
height,  and  this  mass  had  the  shape  of  the  roof  of  a  house  of 
indefinite  length,  then  the  Cordilleras  would  produce  at  a  only 
\  the  effect  of  the  entire  spherical  shell,  as  can  he  easily  proved, 
but  there  are  further  additions  to  be  made  in  order  to  give  a 
more  accurate  idea  of  the  Cordilleras  of  Peru.  The  base  EE  is 
80  or  100  times  greater  than  the  height  \((,  which  augments 
the  effect  in  precisely  the  same  ratio  as  the  angle  at  <(  is 
greater.  This  angle  is  oidy  90°  when  we  find  the  effect  \  of 
that  which  the  whole  spherical  layer  would  produce,  but  on 
account  of  the  great  width  of  the  base  of  the  Cordilleras  the 
angle  is  nearer  170°,  which  doubles  the  effect.  Moreover,  the 
Cordilleras  do  not  terminate  at  the  height  of  Quito  in  a  single 
summit  like  the  ridge  of  a  house  ;  it  is,  on  the  contrary,  quite 
10  or  I'i  leagues  broad  there.  One  can  suppose  then,  without 
fear  of  mistake,  that  the  effect  is  the  greatest  which  can  be 
produced  by  a  chain  of  mountains.  It  is  the  ^  of  that  which  a 
spherical  layer  would  produce,  or  |//^,  and  if  we  add  to  it  the 
attraction  (r— 2/()A,  which  the  globe  ADD  produces  at  a,  we 
shall  have  ( 7'-2h)A  4-  pic*  as  the  expression  for  the  attraction 
at  Quito,  when  rA  expresses  that  at  sea-level. 

49.  The  differciice  between  the  two  is  2//A  —  |//2,  which 
furnishes  the  subject  of  divers  quite  curious  remarks.  If  the 
matter  of  the  Cordilleras  were  more  compact  than  that  of  the 
average  of  the  whole  earth,  and  their  densities  were  as  4  :  3, 
the  difference  2/iA  —  |//^  would  become  zero,  and  the  attrac- 
tion at  Quito  would  be  the  same  as  at  sea-level.  If  the  density 
h  were  still  greater,  our  expression  In-  the  diminution  would 
change  sign  and  become  an  increase,  so. that  the  pendulum 
would  be  longer  at  Quito  than  at  sea-level.     But  it  is  evident 

*  [This  formula  is  independently  foundby  D' Aleinhert  {V3,  vol.'  6,  pp.85-9^i). 
by  Young  (51  and  95,  vol.  2,  p.  27),  and  by  Poixaon  (65.  vol.  1,  pp.  492-6). 

Under  tJie  form  g*=  ffo  {^~—    +    oxJ  *'  ***  known  as  "Dr.  Youitg'-^ 

Rule,"  ichere  ,9'  is  the  value  of  gravity  at  height  h,  and  g^  is  the  valuj  at  the 
sea-level.  Faye  (147)  contends  that  the  last  term  of  the  equation  should  be  left 
out ;  and  if  Airy's  "flotation  theory"  (94),  oi'  Faye's  compensation  theory 
(146 J),  be  true,  there  is  no  doubt  that  this  term  reqvires  correction.] 

31 


fftia 


1;; 


I' 

0 
■I  H 


■1*  ^ji 


-  I 


MEMO  I  Its    ON 


it 


il.i 


tliiit  tliiii[,'s}iro  not  so.  Tlie  difTorenoe  in  the  huijiftli  of  the  pend- 
ulum is  siiflicicMitly  grent  to  lot  us  see  that  tlio  density  of  tlie 
matter  of  wliich  tlio  Cordilleras  is  formed  is  much  smaller  than 
that  of  the  rest  of  the  ^lobe. 

50.  We  have  found  by  experiment  a  diminution  of  a  -Y^jth 
part  in  the  length  of  the  pendulum,  or  in  the  attraction,  as  we 
go  from  the  sea-level  to  Quito.  So  jj'jy  corresponds  to  2AA  — 
|/t^,   as  compared  with  rA,  which  expnssses  the  attraction  at 

sea-level ;  that  is,  we  have  — r.—  = ? —     If  we  put  -  = 


l.'ilil 


r  A 


,..,,, y    which  is  the  ratio  of  the  height  of  Quito  to  the  radius 

of  the  earth,  we  shall  have  ~  =  ^i^  x  '^-^-fc.     Whence 

850 


i3;u     '^'^:i7 


we  deduce  5  =  .TTwTi  ^'  which   tells  us  that  the  Cordilleras   of 

Peru,  in  spite  of  all  the  minerals  they  contain,  have  less  than  ^ 
the  density  of  the  interior  of  the  earth.* 

51.  We  admit  that  this  determination  may  contain  a  few  er- 
rors on  u<icount  of  the  large  number  of  elements  we  had  to  em- 
ploy in  order  to  arrive  at  it.  Nevertheless,  if  we  once  admit 
that  the  attraction,  when  the  other  circumstances  are  the  same, 
follows  exactly  the  direct  ratio  of  the  masses,  we  cannot  doubt 
that  the  Cordilleras  of  Peru  have  a  density  considerably  less 
than  that  of  the  rest  of  the  globe.  If  we  suppose  A  and  5  equal, 
our  expression  for  the  difference  of  the  attractions  at  Quito  and 

at  sea-level  would  become  -—  A ;  which  would  make  the  differ- 

2;- 

ence  between  the  lengths  of  the  pendulum  4  times  too  small,  or 
the  attractions  as  the  square  roots,  instead  of  the  squares,  of 
the  distances  from  the  centre  of  the  earth.  The  attraction  at 
Quito  would  be  less  than  at  sea-level  by  only  the  ^^-Vf^^^  part, 
and  the  pendulum  would  be  really  shorter  by  only  9  or  10  hun- 
dredths of  a  line,  and  in  appearance  by  2  or  3,  an  account  of  the 
different  constitution  and  temperature  of  the  air.  The  differ- 
ence of  the  lengths  of  the  pendulum  is  certainly  greater.  Thus 
it  is  necessary  to  admit  that  the  earth  is  much  more  compact 

*  r^''  .9^^^  Bouffuer's  result  more  accurately,  the  density  of  the  earth  is  4.7 
tinws  that  of  the  Cordilleras.     Saif/ey  (74,  p.  149)  ?ias  made  a  recalculation  of 
tliese  resultti,  with  the  vro)y;r  reduction  to  vacuo,  and  finds  4.25.     He  has  done 
the  same  for  de  la  Condamine's  pendulum  experiments,  with  a  result  4.50. 
For  Addendum^  see  p.  160.] 

82 


11 


THE    LAWS    OK    (UiAVlTATluN 

below  than  above,  and  in  tlie  iiitorior  than  at  tlie  surface.  For 
the  soil  of  Quito  is  like  that  of  all  other  eountries  ;  it  is  a  mixt- 
ure of  earth  and  stones,  with  some  metallic  constituents.  .  .  . 
'i'hose  physicists  who  imagined  a  great  void  in  the  middle  of 
the  earth,  and  who  would  have  us  walk  on  a  kiml  of  very  thin 
crust,  can  think  so  no  longer.  We  can  make  nearly  the  same 
objection  to  Woodward's  theory  oi  great  masses  of  water  in  the 
interior.  But  let  us  continue  to  limit  Oiirselves  to  the  facts,  or 
to  the  only  immediate  deductions  which  we  can  draw  from  them. 
These  deductions  are  confirmed  by  the  observations  described 
in  the  next  chapter,  which  is  in  the  form  1  gave  it  in  Peru  be- 
fore forwarding  it  to  France. 


IV 

MKMOIR    ON    ATTRACTION    AND    ON    TIIP]    MANNER   OF    ORSERV- 

INO    WHETHER    MOUNTAINS    EXIIIMIT    IT    (rEAI)    AT   THE 

ACADEMIE    DES    SCIENCES,   IN    OCTOBER,  Vi'.V.i) 

52.  It  is  very  difficult  not  to  accept  attraction  as  a  principle 
of  fact  or  of  experience.  The  most  rigid  Cartesians,  like  all 
other  philosophers,  cannot  dispense  with  it  in  this  sense.  All 
they  can  do  is  to  reserve  to  themselves  the  right  of  explaining 
it.  .  .  .  Since  all  the  planets  circle  about  the  sun,  there  must 
necessarily  be  a  force,  I  shall  not  say  shoving  them  or  drawing 
them,  but  rather  transporting  them  at  each  moment  towards 
this  star.  .  .  .  Nothing  prevents  us  from  giving  to  this  force 
the  name  **  attraction,"  and  from  trying  to  assign  to  it  a  phys- 
ical cause.  .  .  . 

[BongueraffinHs  fhat  in  esfahh',sJtinf/  a  nem  principle,  it  is  not 
onlif  necessary  to  prove  the  insufficiency  of  all  others,  bnt  their 
impossibility  also.  ] 

54.  While  waiting  for  all  this  to  happen,  it  will  contribuce 
to  the  perfection  of  physics  if  we  examine  more  carefully  into 
attraction  as  a  fact  taught  by  experience.  ...  It  appeared 
to  me  that  if  all  bodies  act  "at  a  distance."  in  proportion 
to  their  mass,  and  according  to  the  other  laws  which  we  know, 
such  enormous  masses  [the  mountains  of  Peru]  should  pro- 
duce a  marked  effect.  I  am  well  aware  that  they  are  very 
small  compared  with  the  whole  earth  ;  but  one  can  approach 
1000  or  2000  times  nearer  their  centre,  and  if  it  is  true  that 
c  88 


,i 


i| 


MEMOIRS    UN 


the  tittruotioiis  increase  not  only  simply  in  tlic  Siimc  ratio  as 
the  (listunoes  (liniinisii,  but  in  the  inverse  ratio  of  their  squares, 
one  ouglit  to  have  a  kind  of  eojupensation. 

55.  I  sliall  content  mysi^lf  with  justifying  tliia  in  the  case 
of  a  singhi  mountain  caMcd  Chimborazo,  the  base  of  wliich 
one  is  oiiligcd  to  j)aHS  in  going  from  tlie  sea-side  at  fruaya- 
fjuil  to  the  more  inhal)ite<i  part  of  tlie  provim^e  of  Quito, 
which  is  enclosed  between  the  two  chains  of  mountains  liere 
formed  by  the  Cordilleras,  whose  distance  aj)art  is  8  or  9 
leagues.  Chimborazo  must  be  .'UOO  or  3200  toises  above  the 
sea-level  [he  aflcrmrrfls  fntoid  if  to  he  3:^17  toises],  and  1700 
or  1800  above  the  level  of  the  plat(^au.  We  know  exa(!tly  the 
relative  heights  of  all  the  mountains  we  have  seen,  but  not 
having  yet  been  able  to  compare  any  one  witli  sea-level,  we 
are  ignorant  of  their  absolute  heights.  Chimborazo  has  roots 
which  extend  very  far  and  become  merged  in  those  of  the 
other  mountains,  so  that  it  is  very  difficult  to  determine  the 
true  extent  of  its  base.  It  must  be  more  than  10,000  or  12,000 
toises  in  diameter.  But  when  we  mount  as  high  as  possible, 
to  where  the  snow  begins,  which  is  850  toises  from  the  top 
and  renders  the  higher  parts  inaccessible,  the  mountain  is  still 
more  than  3500  toises  in  diameter.  The  top,  instead  of  ter- 
minating in  a  point,  is  rounded  and  blunt,  and  appears  from 
below  to  have  a  width  of  300  or  400  toises.  From  these  di- 
mensions one  can  estimate  its  huge  mass.  In  the  present 
investigation  we  need  to  know  its  height  above  ground  only, 
not  above  sea-level.  Even  so  it  must  be  20,000,000,000  cubic 
toises  in  volume.  This  is  about  the  T.ToTn^oo.ooo^'^  P^^*-  ^"^7 
of  the  globe,  and  the  effect  of  the  attraction  would  be  absol- 
utely insensible,  if  one  considered  the  quantity  of  matter 
only.  But  as  we  can  place  ourselves  at  1700  or  1800  toises 
from  the  centre  of  gravity  of  the  mountain,  or  1900  times 
nearer  to  it  than  to  the  centre  of  the  earth,  this  proximity 
ought  to  increase  the  effect  about  3,000,000  times,  and  so 
make  it  about  2000  times  less  than  that  which  gravitation 
produces,  or  the  attraction  caused  by  the  whole  mass  of  the 
earth.  This  we  get  by  employing  only  a  rough  calculation 
and  the  lowest  estimates.  Calling  the  action  of  the  moun- 
tain 1,  and  that  of  the  earth  2000,  the  direction  of  attraction 
should  be  deflected  from  the  vertical  by  about  1'  43".  A 
plumb-line  which  would  be  directed  exactly  to  the  centre  of 

84 


rill!:    LAWS    <M'     (illA  V  ITATloN 


the  earth,  if  its  jiiiias  \vor«(  ex|)(>se(l  to  the  earth's  attruftioii 
alone,  oiij^ht  tlioii,  o!i  aoconnt  of  the  action  of  tiie  mountain, 
to  ho  incliiUMl  l)y  tiiis  sjiine  fjuantity,  whioii  is,  as  we  see,  (juite 
c!onsi<]orahh». 

50.  lUit  how  ran  wo  recognize  tiii«  inclination;  for  all 
gravitating?  hodies  must  ho  ('(jually  suhject  to  it,  and  wo 
seem  to  lack  a  term  of  comparison?  It  would  he  useless  to 
have  recourse  to  the  level  surfa(!es  of  the  lieaviest  lirjuids, 
since  the  attraction  heinjjf  equally  altered  with  respect  to 
them,  their  surfaces,  instead  of  heing  jjorfectly  horizontal, 
must  sutler  the  siimo  incliiuition.  We  see  plainly,  then,  that, 
in  order  to  judge  of  the  amount  of  this  alteration,  it  will  he 
of  no  use  to  look  just  about  us,  we  must  seek  another  ver- 
tical line  far  oil'  which  is  subject  to  no  action  from  the 
mountain.  Hut  again,  how  are  we  to  compare  one  vertical 
with  another;  or  measure  the  an<de  which  they  make  in 
meeting  towards  the  centre  of  the  earth,  and  that  with  suf- 
Hcient  accuracy?  If  while  on  the  mountain,  we  observe  with 
the  quadrant  the  height  of  a  point  far  (^tf,  and  then  go  to 
that  point  and  measure  the  height  of  the  former  place,  it  is 
true  that  by  the  dilference  of  these  two  heights  we  can  judge 
of  the  relative  positions  of  the  two  vertical  lines.  Hut  be- 
sides that  we  must  know  the  exact  distance  from  one  to  the 
other,  it  will  be  necessary  also  to  suppose  that  the  visual  ray 
is  a  straight  line  ;  and  it  is  not  oidy  certain  that  this  is  not 
true,  we  know  that  it  is  subject  by  refraction  to  a  very  ir- 
regular curvature.  We  cai:not  determine  this  (curvature  with 
sufficient  exactness  to  enable  us  to  find  the  effect  of  *Jie  at- 
traction. It  seems  to  mc,  therefore,  that  we  must  seek  in 
the  heavens  a  term  of  comparison.  By  this  means,  however, 
we  shall  easily  overcome  every  difficulty;  and  what  a  moment 
ago  seemed  an  impossibility  becomes  at  once  very  simple. 

57.  We  have  but  to  station  ourselves  to  the  north  or  to 
the  south  of  a  mountain,  and  as  near  as  possible  to  its  centre 
of  gravity,  and  observe  the  latitude.  This  observation  can  bt^ 
made  with  the  greatest  accuracy  oidy  by  using  a  quadrant 
or  other  equivalent  instrument  whose  plumb-line  will  be  de- 
flected toward  the  mountain;  this  is  the  same  as  saying  that 
the  zenith  will  recede  from  the  mountain.  Then  we  must 
go  east  or  west  of  this  station  to  such  a  distance  that  the  at- 
traction is  negligible;  and  if  we  observe  the  latitude  in  this 

35 


■. 


' 


I 


I- 


mi:  MO  IKS    ON 


il  I ,''  I 


HPcohd  \)\inv.  with  tlu?  saiiio  Ciiro  siiid  with  thn  same  iiioans 
UH  ill  tlic  first,  it  is  ovidciit  tiiiit  all  tlic  (lifTcroiicc  which  wo 
Hhall  ohsiirvo  will  In;  due  to  attraction.  In  onit-r  to  have 
tiiiH  second  .station  precisely  east  or  west  of  the  first,  we  must 
ohserve  the  azimuth  of  the  sun  at  its  rising;  or  setting,  hy 
findiji^  its  position  with  reference  to  some  easily  distinguished 
point  on  the  horizon  ;  in  doing  so  we  must  often  suppose  the 
latituile  known;  hut  the  error  we  nuiy  make  on  this  supposi- 
tion will  he  of  no  consefjuence,  and  it  will  always  he  easy  to 
find  two  stations  on  tlie  same  parallel  of  latitude  to  v/ithin 
',]  or  4  sixtieths  of  a  second.  The  latitude  will  he  found  pre- 
cisely the  same  in  the  two  places,  if  the  vertical  line  has  not 
heen  altered  in  the  first.  Suppose,  however,  that  without 
seeking  the  latitude,  we  ohserve  simply  the  meridian  alti- 
tudes of  a  star  at  the  two  stations  ;  the  dilTerence  of  these 
two  altitudes  will  indicate  equally  well  the  deflection  of  the 
vertical  line.  It  is  evident  that  all  the  stars  which  pass  the 
meridian  on  the  side  of  the  apparent  vei'tical  line  next  to  the 
mountain  will  appear  lower  at  the  first  station  than  at  the 
second  ;  for  as  the  plumh-line  approaches  the  mountain  the 
apparent  zenith  recedes  from  it  and  from  these  stars.  It  will 
be  quite  the  reverse  with  those  stars  which  pass  the  meridian 
on  the  other  side  of  the  apparent  vertical  line :  they  will  ap- 
pear higher  at  the  first  station.* 

58.  Instead  of  taking  the  stations  both  to  the  north  or  both 
to  the  south,  we  could  take  them  one  to  the  north  and  the 
other  to  the  south,  and  exactly  on  the  same  meridian ;  then 
the  effect  of  the  attraction  would  be  doubled,  roughly  speak- 
ing, and  we  should  find  the  sum  of  the  contrary  attractions. 
The  vertical  line  would  be  inclined  in  opposite  directions  at 
the  two  stations  ;  and  the  altitudes  of  stars  which  would  be 
increased  in  the  one  would  be  decreased  in  the  other.  The 
physical  effect  being  doubled  would  be  more  sensible,  and 
more  susceptible  of  observation.  If  the  two  points  were 
equally  distant  from  the  centre  of  gravity  of  the  mountain, 
the  action  would  be  equal  at  both,  and  in  order  to  get  each 


*{_I7m  metJiod  of  doubling  the  deflection  caused  by  tJte  mountain,  hy  observ- 
ing not  one  star,  but  at  least  two,  one  north  and  one  south  of  the  statiotuj, 
is  due  to  de  la  Condamine.  See  his  account  of  tJte  expedition  (8,  p.  68),  Zach 
(49)  and  Poynting  (185,  p.  14).  This  is  the  method  actually  employed  by 
BouguerJ] 

m 


THK    LAWS    OK    (',  li  A  V  ITATIoN 


^T^ 


of  tliem  wo  should  hsive  merely  to  tiike  hjilf  of  tlie  (jUiintity 
furnished  hy  the  coiiiparison  of  the  ohserviitions.  In  other 
ciiscH  the  division  wonhl  hv  ii  little  more  ditlienlt;  neverthe- 
less it  would  he  sunieient,  Jis  we  sluill  shew  later,  to  divi<ie 
tljo  sum  of  the  eontniry  attnietions  pronortionaliy  to  the  pro- 
ducts of  the  f|uantity  hy  which  each  station  is  more  north  or 
more  south,  respectively,  than  the  centre  of  ^'ravity  of  the 
mountain  and  the  cu])e  of  the  distance  of  the  other  station, 
respectively,  from  the  sumo  centre.  Thus  W((  are  untler  the 
necessity  of  knowin^j  the  situation  of  each  station  with  refer- 
ence to  the  mountain  ;  hut  w((  must  know  tin*  distance  from 
one  station  Jji)  the  other  also,  in  order  to  determines  j^'eometric- 
ally  the  difference  of  latitude  hetween  them.  It  is  evident 
that  this  difference'  must  itself  j)roduce  a  change  in  the  alti- 
tude of  each  star,  and  we  must  know  it  hefore  we  can  tell  what 
is  the  douhle  effect  of  the  attracticMi.  To  ohtain  the  difference 
in  latitude  of  the  two  places,  it  would  suffice  ordinarily  to 
measure  to  the  east  or  to  the  west  of  the  mountain  a  hase 
directed  nearly  north  and  south,  and  to  form  on  this  hase  two 
triangles  which  end  at  the  two  stations. 

50.  This  way  of  makinij  two  ohservations  from  different 
sides  of  the  same  mountain  in  order  to  render  the  effect  of  the 
attraction  more  sensihle,  seems  to  me  tiie  more  useful  method, 
as  it  depends  less  on  the  peculiarities  of  the  places.  We  can 
sotnetimes  double  the  effect  also  by  making  the  first  observa- 
tion at  the  north  of  one  mountain  and  the  second  at  the 
south  of  another.  If  the  two  stations  are  not  exactly  on  the 
same  east  and  west  line,  we  have  only  to  determine  geometric- 
ally their  difference  of  latitude,  and  take  account  of  it  in  the 
comparison  of  the  altitudes  of  the  stars. 

GO.  Finally,  it  is  not  only  by  observations  made  at  the  north 
or  at  the  south  that  we  can  discover  whether  mountains  are 
capable  of  acting  "  at  a  distance"  ;  it  can  be  done  also  by  ob- 
servations made  at  the  east  or  the  west;  but  with  this  differ- 
ence, that  it  will  be  no  longer  a  question  of  observing  latitude, 
or  of  taking  the  meridian  altitudes  of  stars;  it  will  be  only  a 
(piestion  of  determining  time  exactly.  It  appears  to  me  that 
this  last  method  would  be  often  preferable  to  the  preceding 
ones,  except  that  it  requires  two  observers.  Suppose  that  the 
first  of  these  is  on  the  east  side  of  a  mountain,  and  the  second 
on  the  west  side  of  another,  or  of  the  same,  mountain.     If  each 

37 


ill 


1  ^1 


; 

t 

i 


ill 


f 


M  i:.M(>i  ks  ON 


of  ihom  ro<^uliit('s  ciircfiilly  u  clironornch'v  hy  (foncspoiulin^  al- 
titudes, it  is  t'vidciit  tliiit  ull  tlu'Ho  iiltihidtts  bLMii«;;  altered  by 
tlio  iittni(;ti()U  \vlii(di  dcilcM'ts  tlie  pliinil)-line,  oacli  (diroiioino- 
tcr  will  l)(!  n><,'iil!ittMl  us  if  the  nieridiim  were  not  exactly  vertic- 
al, but  in(diMed  below  toward  the  mountain,  and  above  uwuy 
from  it.  Let  us  sup|)ose  that  the  attraction  amounts  to  a  min- 
ut(t  of  arc.  and  that  the  two  mountains  uro  un  the  equator  ; 
th(i  lirst  (dironometer  will  denote  midday  4  seconds  of  time 
too  soon,  and  the  otluM'  4  seconds  too  late.  TIjus,  nej;leet- 
iiii,'  tin;  dilTerenco  of  lon«j^itude,  which  we  could  easily  find  by 
measuriui:;  tri^onometrically  the  distance  of  the  two  observ- 
ers apart  and  reducing  this  distance  to  decrees  and  min- 
utes, there  would  be  a  ditTerence  of  8  seconds  of  time  be- 
tween the  two  chronometers.  If  the  two  mountains  instead 
of  being  on  the  equator  were  at  latitude  (10°,  each  minute  of 
imdijuition  whicdi  the  attraction  produced  in  a  plumb-line 
would  produce  8  seconds  of  dilTerenco  in  the  time  of  mid- 
day, and  tluM'efore  HJ  seconds  difference  in  the  chronometers. 
Finally,  to  judju^e  of  the  iittraction  we  need  only  know  the  exac^t 
difference  between  the  (dironometers;  and  to  find  this,  it  would 
always  b(^  sufficient  to  agree  upon  a  signal,  })y  fire  or  other- 
wise ;  and  to  observe  at  both  stations  the  minute  and  second 
of  the  instantaneous  ajjpearance  of  this  signal. 

(11.  I  return  to  tlu;  first  method  because  it  appears  to  me  to 
be  the  simplest ;  that  is,  suppose  we  station  ourselves  always 
to  the  nortli  or  to  the  south  of  the  mountain  and  confine  our- 
selves to  observations  of  the  latitude.  It  is  evident  that  if  we 
take  at  each  station  the  meridian  altitude  of  one  star  only, 
we  must  know  to  the  last  degree  of  nicety  the  condition  of  the 
(juadrunt  we  are  using.  There  is  no  lack  of  methods  for  veri- 
fying this  instrument,  but  there  is  one  which  is  extremely 
valuable  in  the  present  instance,  because,  at  the  same  time  as 
we  work  at  verifying  the  quadrant,  we  are  making  the  observ- 
ations which  decide  the  question  at  issue  ;  and  in  thus  abridg- 
ing the  operations  we  avoid  opportunities  for  errors.  This 
method  is  to  take  the  meridian  altitudes  of  an  equal  number 
of  stars  toward  the  north  and  toward  the  south,  and,  provided 
that  the  state  of  the  instrument  does  not  vary  from  one  observ- 
ation to  another,  it  does  not  matter  if  it  does  change  from 
day  to  d;iy.  If  it  makes  the  altitudes  of  the  stars  on  one  side 
the  zenith  too  great,  it  will  produce  the  same  effect  with  re- 

88 


Til  K    LAWS    OK    iillAVIT  ATION 


i« 


Hpect  to  tlioso  on  tho  otlior  si«l('.  Tlin?*  tlio  cliimjjo  will  inflii- 
niro  only  tlic  sum  of  the  jiltitrnh's  or  llic  coniplcnn'iits  of  the 
jiltitucU'S,  anil  will  not  alter  tlir  tlilTcri'iici'  of  t  he  alt itndcs  taken 
on  the  (lilTerent  sides.  The  jit  tract  ion,  on  the  ('(uitrary,  will  n(»t 
alter  the  sum,  hut  will  ('han;^'i'  the  dilTerenee;  heeanse  at  the 
name  time  that  it  makes  the  stars  on  one  sidt^  too  hi^h,  it  makes 
those  ofj  the  other  side  too  low.  It  will  always  ho  easy  to  sep- 
arate these  two  eaiises.  and  we  shall  not  attrihnto  to  the  one 
that  whieh  arises  from  tln^other.  To  ohtain  at  one  stroke  the 
efTect  of  attraction  without  heini,'  ohli;r(.d  to  know  the  state  of 
tho  quadrant  or  tlu^  decdinations  (d'  the  stars,  we  need  only  ex- 
amine whetlnu"  the  ditTereiices  of  the  meridian  altilndes  taken 
towards  the  north  and  towards  tlu;  south  ar(^  the  same  at  the 
two  stations,  or  whether  they  an^  suhject  to  a  seeotul  difT(!r- 
ence.  lint  it  is  necessary  to  remark  that  the  altitudes  hein^' 
incM'eased  on  tho  one  side  while  they  are  diminisluMl  on  tin; 
other,  it  is  the  half  of  this  S(!cond  dilTc^'once  whi.  h  diMiotes 
the  physical  etToct  of  tho  attraction,  hoth  when  this  elTect  is 
single  and  when  it  is  douhle.  In  this  lattor  case,  it  will  he 
necessary  to  divide  the  total  etTcct  in  tluf  ratio  wITudi  the  sep- 
arate effe(3ts  ought  to  have. 

[/ioHf/nrr  then  proirs  tliix  rafio  in  he  thai  \nvnt\uned  nlnire  [p. 
W'i)  He  adnnts  that  some  nntuiitaiiis  niiiftit  shein  tesx  dttrartion 
than  that  rcf/aired  fit/  .\Vv/7o//'.s'  tfttn  {ar  eren  none),  due  to  the 
existence  of  great  rarities  in  the  tntiss.  He  discusses  the  dijf'erent 
nnn(nt(fins  in  the  neit/hbtHirhood  of  Quito,  and  for  various  reasons 
decides  upon  Chiniborazo  as  the  one  most  suit alAe  for  tlie  experi- 
ment.^ 

Examination  of  the  Attraction  of  Ciiimborazo 

65.  I  did  not  ascend  this  mountain  alone  as  \  did  the  pre- 
ceding one.  I  had  some  time  hefon^  communicated  my  design 
and  all  my  views  to  M.  de  la  Condaminc,  and  when  on  the 
point  of  carrying  them  out  I  mentioned  tlu'm  to  M.  de  Ulloa, 
one  of  tho  two  naval  lieutenants  who  had  assisted  in  the  oh- 
servations  both  of  myself  and  of  i\r.  de  la  Condamine  ever  since 
our  arrival  In  the  domains  of  His  Catholic  Majesty.  These 
gentlemen  obligingly  offered  to  accom})any  me,  not  oidy  in  the 
preparatory  examination,  but  also  during  the  stay  it  was  necess- 
ary to  make  on  the  mountain  side  ;  and  as  I  knew  it  would  be 
to  the  advantage  of  the  observations,  I  hastened  to  accept  the 

80 


MKMOIKS    ON 


P 


'       ill 


ofT**!'.  I  IukI  alivudy  tlioii^^lit  tlitit  ('liiFiiboruzo  fiiHillcil  p- 
proxitniiti'ly  tlu;  lu't'osHitry  conilitioris  :  I  knew  tlitit  it  wii  .y 
niwy  of  iKMM^ss  ;  it  (M)iiI(I  be  Htutn  from  (^iiih),  or  rather  from 
IMcliiiicliii,  from  wliicii  it  wuh  Tr>,<MM)  t(»is(>8  ilirttuiit ;  and  I  liud 

ulrciidy  mi'iiHiirtMl  its  lu'i;,dit On  DiMutmbcr  4tli  wo  ('stub- 

lisht'd  oiirHclvt.'K  on  the  Hotitli  Hidu  of  the  mountain,  at  the  bot- 
tom of  tbo  Hnow  line,  S'V.i  toi.ses  lutlow  tlu;  summit,  but  about 
•J40()  abovo  sni-levi'l,  ami  exacitly  IMO  toises  above  the  pbuso  tit 
Quito  wliero  I  have  always  made  my  observations,  and  IJ44 
toises  al)ove  that  part  of  IMeliiiudia  wliero  tlioro  i8  a  (M'okh 
wliicdi  (;an  be  stieii  from  all  [uirts  »»•■*  the  eity,  and  where  I  passed 
some  days  in  Mareli,  1137,  in  order  to  observer  the  astronomitial 
refraction.  I  shall  not  sptMik  of  the(U)ld  and  the  other  diseuni- 
forts  we  ha<l  to  put  up  with  ;  snow  covered  our  tent  and  all 
the  j^round  around  as  far  as  SOO  or  IMKI  toises  below  us,  and  we 
lived  ill  fear  of  beinj;  buried  undtu'  its  wei«;ht.  It  needecl  con- 
tinual vi<(ilance  in  onler  to  avoid  it.  [M.  dc  intotf  fell  ill,  and 
had  lo  di'sreml  the  tnininhihi  on  Dcmnhcr  \'yfh. ) 

[  Li'/l  alone,  /ioNt/Hcr  and  dv  la  ('Onditniinv  ohsrrird  the  alii- 
fades  of  10  sfars,  4  on  the  sindh  side  and  (5  on  the  north.  The 
fidlotrinf/  are  the  altitudes  as  ajf'erted  /»//  the  error  of  the  instrn- 
nieat  and  hi/  refraction;  ttiej/  arc  the  means  of  t lie  rvadinijs  of  the 
two  ot)ser vers.  \ 


On  tlio  norlli  side  : 

Ciipellii 

First  head  of  Gemini. 

Second  heud  of  Geiniii 

Second  horn  of  Aries. 

First  horn  of  Aries. . . 

Aldehariiii 

On  the  soutli  side  : 

Acarnar 

Canopus 

Tail  of  Cetus 

Siriiis 


Moridiiui  HllitnJoH 

ul  the  flrnt  Htation 

On  14tli 

•w. 

On  15tli  Dnc. 

O 

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40 

68.  Wo  observed  the  meridian  altitude  of  the  sun  three 
times.  De  la  Condamino  found  it  at  the  lower  edge  on  De- 
cember 15th  to  be  07°  54'  ^(5'^  This  altitude,  wliich  is  cor- 
rected for  the  error  of  the  instrument,  but  not  for  refraction 

40 


'FBI 


Til  K    LAWS    OK    rnt  A  VITAT  ION 


ii!ul  punillux,  ^ivos  T  'Vy  M**  Koiitli  for  tlio  liititmlo  of  the 
l»lin'(5  wlicrt!  we  wwr.  I  ulist-rvrd  it  on  tlio  ."»tli  uikI  l\itli  ;  on 
tlir  .Mil  \vt!  hail  not  yrt  rc^^MilattMl  tlu'  rhronofnctor  nor  tnirtMJ 
tlie  meridian,  and  I  t'onnd  1  '  iio'  10".  On  tliu  I'^tli  I  oIihoi'vimI 
llic  apparent  attitiidi^  of  tlio  lower  udgo  of  the  8iin  to  be 
IIS"  :)':J4";  whieh  ^ivert  I"  'M)'  0". 

iWK  As  soon  as  wo  wore  estal)lislio(l  on  ('hirnhorazo,  \  lui<I  sent 
a  tent  al)ont  a  league  and  a  half  to  tlio  west  to  a  place  (billed 
rAi'viml  to  servo  as  the  seeonti  station,  [/iotnfm'r  then  dcscriht's 
the  nwnsHirnii'itls  inadr  tn  Jind  Iki'  cxdrl  position  of  the  scrond 
station  :  it  nuts  '.i'ui)  ,'oisrs  distant  from  t  ft  e  first,  1T4  toiscs  fotrrr, 
and  sonivwhitt  snutli  nf  trrsf  of  it.  Tlifif  tnyan  tttrir  otiscrrtttions 
from  tin'  strond  stdlion  on  Dvc.  IVtth.  Ilcrv  tln'if  snffcrvd  niorr 
front  tlic  wind  and  cold  tlian  at  tlie  more  derated  station,  as  tlieif 
were  more  exposed  to  the  prevailint/  east  wind.  It  filled  their 
eifes  with  dust  and  rontinaal/tf  threatened  to  overturn  the  tents.* 
The  screws  of  the  t/aadrant  ronld  not  tte  tamed  at  ni(jht  withant 
applffint/  heat  to  them..  Then  follows  a  talde  of  the  altitades  as 
of/serred.  Sinre  the  second  station  was  505  toises,  or  W^",  south 
of  the  first,  we  mast  increase  hij  Wl"  all  the  altitudes  of  stars  ob- 
served toward  the  north,  and  diminish  the  others  by  tlie  same 
amount.  Moreover  since  the  second  station  was  lower  than  the 
first  by  174  toises,  the  altitudes  observed  at  the  second  station 
must  be  diminished,  to  reduce  them  to  the  level  of  the  first,  on  ac- 
count of  the  cxces.s  of  astronomical  refraction.  The  folio wimj 
are  the  corrected  altitudes,  and  the  differences  for  each  star  of 
the  mean  determinations  at  the  two  stations  .•] 


I*, 

^1 


^  i\ 


*  [For  de  la  Condnmine'H  account  of  (tie  experiments,  see  his  Journal  (8, 
/).  69  and  8h  pt.  2,  p.  146).] 

41 


II  il 


\ 


\. 


IT 


MEMOIRS    ON 


CORRBCTED  ALTITI'DKH   AT  TIIK   SkCOND   STATION 


Oil  the  norih  side: 

(Jnpella 

First  liead  of  Gemini . . . 

Second  head  of  Omiiiii. 

Second  horn  of  Aries.. . 

First  horn  of  Aries 

Aldelmraii 

On  the  soulii  side 

Acarnar 

Canopiis 

Tail  of  Cetus 

Sirius 


On  2lBt  Doc. 

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17 

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11 

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33 

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33 

56 

38 

38 

57 

36 

73 

4 

50 

75 

8 

n 

F.XCCHH    (>r    ItltitlKtOH 

nt  Iho  llrHt  HlHlioii 
ovor  thoHC  at  iIh 
BGcoiiil,  iifior  lilt' 
liitler  hiivo  bec" 
corroded 


1  24 


1   25 


10 
6* 


1   37i 
1   28 
1    48 

1    33* 


III 


':J  ' 


t  ! 


[Bovgtier  considers  the  observations  on  the  tail  of  Vetns  and 
the  first  horn  of  Aries  as  the  best,  but  thinks  it  most  legitimate 
to  take  the  mean  of  ttie  altitudes  of  each  star  at  each  station  and 
to  give  equal  tveight  to  their  differences.  Tttese  differences  are 
given  in  the  last  column  of  the  preceding  table,  and  are,  lie  main- 
tains, too  large  and  too  uniform  to  be  due  to  any  defect  in  the  ob- 
servations. Tlie  averages  of  the  excess  of  all  the  stars  on  the 
north  side  and  all  those  on  the  south  side  are  now  to  be  taken.] 

74.  .  .  .  Tliey  Jjive  about  1'  19"  us  the  mean  excess  for  the 
north  stars,  and  1  34"  for  the  south.  The  second  difference  is 
15".  I  leave  it  to  my  readers  to  say  whether  such  a  quantity 
is  sufficiently  established  by  the  means  employed.  My  quad- 
rant was  2.5  ft.  in  radius,  and  it  must  be  remarked  that  any 
errors  which  may  exist  in  its  graduation  are  of  no  importance 
here  ;  since  we  have  to  do  not  with  the  altitudes  tliemselves, 
but  with  their  differences.  Suppose  we  admit  the  15",  it  will 
^\\Q  7".5  for  the  effect  of  attraction;  it  would  be  much  greater 
if  we  compared  the  tail  of  Cetus  with  the  first  horn  of  Aries. 
However  this  is  not  the  complete  and  absolute  effect ;  for  if 
J^ttraction  really  takes  place,  the  mountain  must  have  some 
effect  at  the  second  station,  which  was  about  4572  toises  from 
the  centre  of  the  mountain,  and  61°. 5  to  the  west  of  south. 
At  the  first  station  we  were  nearly  16°  west  of  south,  and  1753 


*  [T/iis  is  evidently  a  misjyrintfor  1'  16".] 
43 


THE    LAWS    OF    GHAVITATION 


toises  distant.  From  tliese  data  we  find  that  tlio  effect  at  the 
iioarer  station  is  to  the  e.*^?ect  we  ouglit  to  tind  at  tiie  otiier  as 
lIJoS  :  100,  '31*  as  13 j\  :  1  nearly,  lint  since  our  observations 
^nve  only  the  difference  of  the  two  effects,  we  must  increases 
T".5  by  a  13tli  or  14th  part  of  itself  in  order  to  have  the  total 
effect  [which  makes  if  <S"J. 

75.  We  must  admit  that  this  effect  is  very  different  from 
what  we  had  expected.  But  we  know  so  little  about  the 
earth's  density,  and  on  the  other  hand  that  of  the  mountain 
may  be  so  different  from  that  which  we  have  assumed  it  to  be, 
that  there  is  no  reason  to  be  surprised  at  anything.*  [//  is  a 
tradition  among  the  natives  that  Chiinborazo  is  an  extinct  vol- 
cano, and,  if  so,  its  density  would  1)e  vtry  hard  to  estimate, 
lionyuer  thinks  it  miyht  be  better  to  experiment  on  smaller  and 
denser  monntains.]  It  is  very  probal)le  that  we  shall  find  in 
France  or  in  p]ngland  some  hill  of  sufficient  size,  especially  if 
we  double  the  effect ;  and  1  shall  be  delighted  if  1  find  on  my 
return  that  the  experiments  that  shall  have  been  made  either 
confirm  mine  or  throw  new  light  on  the  matter.  [At  Kiobamba 
in  Peru,  December  30,  1738.] 

[In  an  appendix  Bouyuer  states  that  after  a  more  thoronyh 
snrvey  of  the  Cordilleras  he  failed  to  find  a  more  satisfactory 
place  at  which  to  repeat  his  experiment.  He  snygests  ttiat  the 
converse  effect  be  experimented  upon  :  viz. ,  the  decrease  in  grav- 
ity due  to  some  deep  canon  amony  mou.Jains.  Assuminy  such 
yreat  cavities  in  Chimborazo  as  would  make  its  real  only  half 
its  apparent  vohwie,  he  finds  his  results  would  make  it  G  or 
7  times \  less  dense  than  the  earth;  this  he  thinks  not  unreason- 


*  [De  la  Condnmine  also  laid  little  stress  on  the  numerical  result  of  tJie  plumb- 
I'ne  ex})eriment ;  foi'  tie  says  {S,  p.  69),  "  if  we  can  deduce  from  it  nothing 
decisively  in  favour  of  t/ie  Newtonian  attraction,  at  least  we  find  not/iing  con- 
trary to  tJtat  theory. "] 

f  [In  a  critical  analysis  of  all  the  experiments  made  todeto'mine  the  density 
of  the  earth  vp  to  t/iat  time,  Saigey  (74,  p.  151  and  in  his  ''Petite  P/iysique  du 
Globe,"  Paris,  1842,  pt.  2,  p.  151),  in  1842,  stated  that  liouguer's  calcnlations 
were  erroneous,  because  tie  confused  llie  centre  of  attraction  with  the  centre  of 
gravity  of  tJie  mountain;  he  refers  to  a  metJiod,  by  means  of  whicJi,  using 
Bouguer's  own  mean  result,  lie  deduced  the  density  of  tlie  earth  to  be  4.62 
times  ttiat  of  Uie  mountain  ;  but  adds  tliat  if  he  had  used  only  the  results  from 
observations  on  tlie  tail  of  Cetus  and  tJie  first  horn  of  Aries,  the  result  would 
tiave  been  1.83,  which  is  almost  exactly  tJie  same  as  ttoit  found  by  MasJcelyne  by 
tlie  same  metliodfor  the  Iiill  S^heliallien.] 

43 


if  I 


M 


t'. 


i  i ; 


-1 
I: 

,4  5 


ii 


II 


MEMOIRS    ON    THE    LAWS    OF    GRAVITATION 

able.  Bonguer  further  remarks  that  the  distribution  of  density/ 
in  the  earth  may  be  such  that  the  maximum  attraction  of  the 
earth  is  not  at  its  surface  but  at  some  J' stance  beneath  «7.] 


[In  connection  with  this  work  of  Bouguer  shoiild  be  read  a 
paper  (9)  presented  by  him  to  the  Academy  on  April  28,  1750, 
on  the  possibility  of  detect irig  the  deflection  of  a  pluuib-line  due 
to  the  ebb  and  flow  of  the  tide. 

Valuable  accounts  and  discussions  of  Bonguer' s  work  in  Peru 
are  given  by  von  Zach  (43,  44,  and  40),  Schmidt  (04,  vol.  2, 
p.  475),  Todhunter  (140.  chap.  12),  Za7iotti- Bianco  (\4S^,  pt. 
2,  pp.  122-25),  and  Poynting  (185,  pp.  10-14.] 


Pierre  Bouguer  was  born  in  1098  at  Oroisic,  Bretagne,  and 
was  educated  at  the  Jesuit  College  at  Vannes.  He  succeeded 
his  father  as  professor  of  hydrography  at  Oroisic  in  1713,  and 
in  1730  accepted  a  similar  position  at  Havre.  His  investiga- 
tions concerning  the  intensity  of  light,  embodied  in  a  work, 
Essai  d'optiqne  sur  la  gradation  de  la  lumiere,  published  in 
1729,  led  to  his  being  elected  a  member  of  the  Academy  of 
Sciences  in  1731 ;  he  was  promoted  to  the  ottico  of  pensioned 
astronomer  in  1735.  Along  with  two  other  members  of  the 
Academy,  MM.  de  la  Condamine  and  Godin,  he  was  sent  to 
Peru  in  1735  to  measure  the  length  of  an  arc  of  the  meridian 
near  the  equator.  Their  labors  there  lasted  ten  years,  and  the 
results  of  their  observations  were  published  in  1749  in  La 
Figure  de  la  Terre,  from  which  we  have  given  the  preceding  ex- 
tracts. For  several  years  afterwards  Bouguer  was  engaged  in 
a  bitter  controversy  with  de  la  Condamine  concerning  their  re- 
spective shares  in  the  Peruvian  researches.  In  addition  to  his 
works  on  photometry,  ho  published  several  valuable  treatises 
on  navigation,  and  various  papers  on  atmospheric  refraction 
and  other  optical  problems,  and  on  mechanics.  He  died  at 
Paris  in  1758. 

44 


45 


i| 


Ml 


THE    BERTIER  CONTROVERSY 


THE    BERTIER   CONTROVERSY 


1 


In  June,  1709,  there  appeared  in  the  Journal  den  Sciences  ef 
lies  Beaux  Arts  a  letter  (11)  by  a  M.  (Joultaud,  wlio  signed 
liimself  Former  Professor  of  Physics  at  Turin.  In  it  he  de- 
scribed some  pendulum  experiments  madc^  in  the  Alps  of  Savoy. 
Ho  claimed  to  have  found  that  at  a  height  of  1085  toises  above 
the  base  of  the  mountain  the  pendulum  gained  28'  in  2  months  ; 
20'  22"  in  'A  months  at  a  height  of  514  toises;  an.l  15'  4"  in  175 
days  at  a  height  of  210  toises.  So  that  it  appeared  as  if  the 
attraction  of  gravitation  increased  with  the  distance  from  the 
earth's  centre,  instead  of  behaving  according  to  the  Newtonian 
law.  A  full  account  of  the  apparatus  and  observations  was 
added,  and  there  seemed  no  reason  why  credence  should  not  be 
given  to  the  results.  The  advocates  of  the  Newtonian  theory 
felt  called  upon  to  account  for  this  phenomenon  consistent- 
ly with  their  doctrine.  D'Alembert  (12  and  13,  vol.  6,  pp. 
85-92)  attacKed  the  problem  and  found  that  the  Newtonian 
theory  was  adequate  to  explain  the  fact,  provided  the  mean 
density  of  the  earth  were  about  three  eighths  of  that  of  the 
mountain.*  An  abstract  of  Coultaud's  alleged  observations  is 
given  by  David  (14). 

In  Dec,  1771,  another  letter  appeared  in  the  same  journal 
(15)  signed  by  one  Mercier,  and  addressed  to  Gessner,  Pro- 
fessor of  Physics  in  the  Univ.  of  Geneva.  It  described  experi- 
ments made  in  Valois  similar  to  those  of  Coultaud  and  with 
similar  results,  namely,  that  the  attraction  of  gravitation  is 
directly  proportional  to  the  square  of  the  distance.  D'Alem- 
bert  then  discussed  the  question  again  (13,  vol.  G,  pp.  93-98). 
Further  explanations  on  the  Newtonian  theory  were  forth- 


^i 


f..^:t 


:U1 


'IIM 


*  Compare  the  conclusion  of  Bout^uer  on  p.  31  of  ibis  volume. 

47' 


MKMUlltS    UN 


II 


n 


ff 


corning  from  Le  Sago  (10)  and  Laliindo  (17).  RoitTe  also  dis- 
cussed the  experirnonts  (18). 

Tliese  results  of  Coultaud  and  Mercier  seem,  however,  to 
have  been  a  cause  of  great  exultation  to  a  certain  number  of 
scientists,  especially  ecclesiastics,  who  contended  that  the  New- 
tonians wished  to  take  from  them  their  Father,  their  God,  in 
asserting  that  bodies  attract  and  move  of  themselves  without 
any  Prime  Mover.  It  is  hard  to  believe  that  this  feeling 
existed  so  late  as  a  century  ago.  One  of  the  most  active  of 
the  opponents  of  the  theory  of  Newton  was  Father  Bertier, 
de  rOratoire,  who  founded  his  4th  volume  of  Les  Prinriprs 
Physiques  on  the  above  experiments.  For  several  years  a  warm 
discussion  raged  among  French  physicists  over  the  question. 

Le  Sage,  having  had  his  suspicions  aroused  by  some  passage 
in  Mercier's  letter,  began  a  careful  investigation  into  the  gen- 
uineness of  the  experiments  both  of  Coultaud  and  Mercier. 
lie  found  them  to  be  fabrications  from  beginning  to  end  (I'J). 
Le  Sage  does  not  mention  whom  he  supposes  to  be  the  j)er- 
petrators  or  instigators  of  the  fraud. 

A  new  impetus  was  given  to  the  discussion  by  the  publica- 
tion of  2  letters  (20)  from  Father  Bertier  describing  experi- 
ments with  the  balanoe  similar  to  those  performed  by  members 
of  the  Royal  Society  of  London  a  century  before  ;  but  Bertier 
writes  as  if  the  idea  were  entirely  a  new  one.  The  length  of 
the  string  used  to  suspend  the  weight  from  one  arm  of  the 
balance,  after  it  had  been  counterpoised  in  the  pan  above,  was 
74  ft.  In  one  case  weights  of  25  lbs.  were  used,  and  when  one 
of  them  was  suspended  at  the  end  of  the  string  it  lost  in  weight 
1  ounce  3.5  drachms.  Bertier  concluded,  much  to  his  satisfac- 
tion, that  bodies  weigh  more  the  farther  they  are  from  the 
centre  of  the  earth.  Roiflfe  followed  with  a  paper  (21)  discuss- 
ing the  experiments  made  thus  far  and  remarking  that  Bertier 
had  not  taken  account  of  the  difference  in  the  density  of  the 
air  at  the  two  levels.  Le  Sage  also  criticised  Bertier  very 
harshly  (22).  Repetitions  of  Bertier's  experiment  were  made 
by  M.  David,  and  Fathers  Cotte  and  Bertier  (23),  the  one  with 
a  string  of  170  ft.  and  weights  of  1220  lbs.,  the  others  with 
a  string  of  45  ft.  and  weights  of  150  lbs.  The  one  reported  a 
loss  in  weight  of  1  oz.,  the  others  of  2  lbs.,  in  the  same  direc- 
tion as  indicated  by  Bertier's  iSrst  experiment.  An  article  by 
David  (24)  in  answer  to  Le  Sage  contains  some  scornful  strict- 

48 


TlIK    LAWS    OK    GRAVITATION 


iiros  on  Nowton  anil  liis  principles  which  form  amusing  read- 
ing at  this  late  date.  Uozier  ('^5)  criticised  all  the  experi- 
ments made  on  the  laws  of  gravitation  ;  he  refers  to  some 
more  made  by  Bertier  (:iG)  frotn  which  the  latter  concluded 
that  the  loss  in  weight  was  proportional  to  the  length  of  stri?ig 
and  to  the  weight.  Uozier  then  announced  the  details  of  some 
experiments  of  a  similar  kind  made  by  himself,  which  gav(f 
quite  discordant  results.  David  wrote  another  letter  ('^7)  with 
more  details  of  his  experiments,  but  adding  nothing  of  value. 
Bertier  followed  with  a  siinilar  letter  ('iS).  A  committee  of 
thfc  Academy  of  Dijon  repeated  the  experiments  with  a  sensit- 
ive balance,  and  found  (2!J)  no  change  in  weight  except  that 
due  to  the  different  densities  of  the  air  at  the  higher  and 
lower  levels.  Some  experiments  on  this  same  subject  were 
made  by  Achard  (IJ3) ;  he  found  by  using  first  a  string  aiid 
then  a  brass  chain  with  which  to  suspend  the  masses,  that  the 
changes  in  weight  of  the  suspended  nuiss  could  be  ascribed  to 
variations  in  the  temperature  and  dampness  of  the  air.  Dolo- 
mieu  (30)  made  some  experiments  with  a  weight  sus])ended  in 
a  mine,  similar  to  those  of  Dr.  Power  (page  2) ;  his  results 
permitted  no  definite  conclusions  as  to  change  in  weight.  In 
connection  with  this  work  we  might  notice  a  valuable  article 
by  Le  Sage  (34)  on  the  history  of  the  theory  of  gravitation  and 
the  experiments  made  concerning  it.  lie  gives  a  brief  account 
of  the  views  of  Gilbert,  15acon,  Kepler,  Beaugrand,  Fermat, 
Pascal,  Roberval,  Descartes,  and  Gassendi  ;  finally  he  dis- 
cusses Dolomieu's  experiments  and  the  possible  variation  in 
density  beneath  the  surface  of  the  earth. 

The  controversy  closes  with  the  appearance  of  a  letter  (30) 
from  Bertier  making  an  humble  retraction  of  his  statements 
regarding  the  deductions  to  be  drawn  from  his  experiment ;  he 
admits  that  such  experiments  do  not  prove  that  bodies  weigh 
more  as  they  are  farther  from  the  earth,  but  he  declines  to  give 
up  his  belief  that  such  is  the  case,  lie  again  inveighs  against 
those  who  "by  means  of  101  different  laws,  which  they  make 
God  create  to  cover  their  ignorance,  explain  everything  with  a 
facility i-hat  is  truly  delightful." 
D  49 


m 

>  i 


i  liFi 


51 


I  i, 


/■,  ii 


THE   SCHEHALLIEN   EXPERIMENT 


.1 


J!  r 


i 


THE   SCHEHALLTEN   EXPERIMENT 


In  1772,  \fjiskolyno,  the  Kii<(lisli  AstroiionuM-  Uoyjil,  pro- 
posed to  tlio  Uoyiil  Sooioty  {'M)  tluit  i\\v  cxporimciit  of  lioii- 
^'uer  on  the  attriictioii  of  a  inoimt.'iiii  ho  repojitcd  in  (Jroat 
Britain,  as  Hoiit^iior  InrnHclf  had  sii«;<ijest(Ml  'M)  years  Ix'foro. 
Maskelyne  liad  heoii  informed  of  two  phiccs  which  iui.i,dit  Im' 
convenient  for  the  purpose.  One  was  near  the  i'onliiies  of 
Yorksliire  and  Lancashire,  on  the  hill  Whernside  ;  the  otiier 
in  (Jund)erland,  on  the  iiiil  Ilelvellyn.  The  proposal  was 
favoural»iy  received  hy  the  Society,  and  Mr.  ('iiarU^s  Mason  was 
sent  to  examine  various  hills  in  Kn<riand  and  Scotland,  and  to 
select  tiie  most  suitahle  (-i'-i).  Mason  found  that  the  two  hills 
referred  to  by  Maskelyne  were  not  suitable;  and  fixed  upon 
Schehallien  in  Perthshire  as  offeriiii^  the  best  situation.  At 
the  earnest  solicitation  of  the  Royal  Society,  Maskelyne  him- 
self undertook  to  make  the  necessarv  observations.  He  had  at 
his  disposal  a  lO-foot  zenith  sector,  and  all  his  other  instru- 
ments were  the  best  of  their  kind  at  the  time.  The  work  was 
begrun  in  the  summer  of  1774.  The  niethod  of  findin<j;  the  de- 
tiection  of  the  plumb-line  due  to  the  hill  was  exactly  the  second 
of  the  methods  described  by  liouguer  (page  JJO) ;  he  took  read- 
ings of  the  zenith-distances  of  certain  stars  at  two  stations,  one 
north  and  one  south  of  the  hill,  aiul  by  this  means  doubled  the 
deflection  of  the  plumb-line.  Between  Juno.'JOth  and  Septem- 
ber 2'^d  he  took  100  star  observations  from  the  south  station, 
and  108  from  the  north  station;  in  all  IV.17  observations  on  43 
stars.  At  the  same  time  a  very  elaborate  survey  by  triangida- 
tion  was  made  of  the  dimensions  and  form  of  the  hill.  This 
was  considered  as  made  up  of  a  very  large  number  of  prisms, 
sufficient  data  for  the  determination  of  each  of  which  were  col- 
lected during  the  survey. 

53 


1 


i  I 


f:f^ 


\  ll 


<•    f  J^i 


MKMOIIIS    ()N 


lii^ 


III  lii.H  \ni\)or  {'.Vi)  di'scrilHii^  tho  operations,  iMuskelyiio  cal- 
(Milutcs,  from  4(>  only  out  of  tlx*  XVi  ohKcrvations,  that  tho  up- 
piirciit  (lilTrri'iicc  of  latitiule  hctwcoii  tlio  two  HtatioiiH  \h  r)4".(».* 
Tho  triio  (lilTrrciuM)  of  latitiulu  is  4;{",  Icaviii^j  11". (}  duo  to  tho 
•  oiitrary  attractions  of  the  hill. 

From  a  rouj^h  cahMilatioii,  assuming  tho  density  of  tlu; 
mountain  to  ho  tho  samo  as  tho  nu>an  density  of  tho  oarth, 
and  that  tho  law  of  attracrtion  is  that  of  tho  invorso  scjuaro 
of  the  distan(M\  Maskolyint  found  that  tho  attra(;tion  should  bo 
twioo  that  found  hy  ohservatiojj.  lionco  tho  moan  density  of 
tho  earth  is  twiet;  that  of  tho  hill.  A  more  exact  calcidation 
was  promised  for  tho  future.  Maskelyno  draws  two  imiin  con- 
clusions: (1),  tiiat  Schohallion  has  an  attraction,  and  so,  there- 
fore, has  every  mountain  ;  {'i),  that  tho  inverse  scjuare  law  of 
till'  distance  is  confirmed  ;  for  if  the  force  wore  only  a  little 
atT(fctiM|  by  the  distan(!o,  the  attraction  of  tho  hill  would  be 
wholly  insensible. 

The  survey  of  the  hill  and  its  environs  was  made  during  the 
years  1774,  1775  and  ITTO.  The  calculation  of  the  attraction 
of  tho  hill  from  thosii  measurements  was  undertaken  by  Mut- 
ton,! who  employed  several  new  and  interostiiiit^  methods.  A 
full  act'ount  will  bo  found  in  his  paper  (l{7  and  47,  vol.  "i, 
pp.  l-(JS).  Assumin*^  that  the  density  of  the  hill  is  the  same 
as  the  mean  density  of  the  earth,  Ilutton  found  that  the  attrac- 
tion of  the  earth  is  to  tho  sum  of  the  contrary  attractions  of 
the  hill  as  iWV,]  :  1.  Now  Maskelyno  had  found  the  deflection 
due  to  the  contrary  attractions  of  the  hill  to  be  IT'.G;  whence 
the  attraction  of  the  oarth  is  to  the  sum  of  the  attractions  of 
the  hill  as  1  :  tan.  ll".tJ,  or  as  17781  :  1 ;  or,  allowing  for  the 


m 


if 

•  f 

J  -I 

,.k. 


*  Von  Zaeli  (49,  App.,  pp.  686-693)  Una  calculated  the  results  from  all 
of  the  H87  observjilifins,  and  finds  for  the  iippureut  difference  of  latitude 
54". 651,  and  for  the  deflection  due  to  the  contrary  attractions  of  tlie  lull 
11". 632  ;  whi(!h  is  in  i'litire  accord  with  Maakelyne's  calculations.  Saigey 
(74,  p  153)  also  subjected  the  result  to  a  test  which  was  satisfactory. 
Zanolti-Bianco  states  (148A,  pt.  3,  p.  134)  that  Saigey  maintained  that 
Maskelyne  did  not  choose  his  station  at  the  most  favourable  part  of  the 
hill-side,  and  that  if  he  had  done  so  he  would  have  found  the  deflection 
14"  inst<'ad  of  11".6. 

f  For  Ilutton 's  own  estimate  of  his  share  in  the  work,  and  for  his  con- 
tempt for  Cavendish's  experinieiu,  see  bibl.  No.  45.  For  a  good  account 
of  Button's  method  of  calculation,  see  Zanotti-Bianco  (148^,  pt.  3,  pp.  126- 
32);  see  also  Helmert  (148,  vol.  3,  pp.  368-80). 

54 


Til  K    LAWS    OK    i.  KAVI'IArinN 


rontrifupjil  forco,  uh  ITH()4  :  I  iHMirly.  I!<mht  thp  iiicuii  donsity 
of  the  eurtli  in  to  tlu»  donHity  of  the  liill  us  1TS(H  :  W.\:\,  or  iih 
1> :  ft  noarly.  Assuming  the  Hp(M'itl(' jjruvity  of  tlio  hill  to  ho 
about  ^.5,  lliitton  ivriiiirks  that,  this  would  ^ivc  4.."»  an  tho 
mean  HpcM'ilic  gravity  of  the  earth.  lliifto?i  revised  this  result 
ill  his  •' Tracts "  (IT,  vol.  'i,  p.  •14);  he  takes  the  sperilic 
jjravity  of  the  hill  as  .'I  and  henco  the  wpeeilic  ^'ravity  of  the 
earth  would  be  5.4  !iearly. 

Play  fair,  with  the  aid  of  liord  Webb  Seymour,  made  a  care- 
ful litholo«;iral  Hurvoy  of  S<diehallieu,  and  published  his  re- 
sultH  in  ISII  (4<>  and  4S).  Ife  found  that  the  hill  was  made 
up  of  two  (tIasHOS  of  rock,  (|uartz  of  specific  ^raviiy  v*.<!.*{!»Si(;, 
and  micaceous  rock,  including  calcareous,  of  specific  jjravity 
'^.H10.*J!>.  Krom  two  suppositions  as  to  the  distribution  of  these 
two  components  in  the  interior  of  the  hill,  usin^  iliitton's  data 
for  the  attraction,  IMayfuir  calculated  the  mean  density  of  the 
earth  to  bo  4.5r)SH»;  and  4.H«H;!l'.>7  respectively.  IMayfair  con- 
sidered the  experiment  on  Suliehallien  so  exact  that  he  took  the 
mean  of  the  above  results,  4.713,  as  the  best  determination  of 
the  mean  density  of  tho  earth. 

Ilutton  prefers  to  take  '>i.77  as  the  mean  of  IMayfair's  deter- 
minations for  tho  density  of  the  hill,  and  the  density  of  the 
earth  as  ^  of  ;i.77,  or  5  nearly.  In  a  paper  j)ublished  in  18:;il 
(52,  53  and  54),  Ilutton  complains  that  his  share  in  the  S(!he- 
hallien  experiment  has  always  been  underestimated  ;  he  ^ives 
a  brief  account  of  tho  observations,  cahuilations  and  results, 
and  considers  5  as  the  most  probable  value  of  the  mean  density 
of  the  earth.  Ho  shews  that  the  Schehallien  experiment  could 
not  bo  made  to  give  tho  same  result  as  that  of  Cavendish,  5.48, 
unless  the  deflection  11". 0  be  diminished  to  about  10". 5  or 
10". 4.  which  is  manifestly  too  great  an  error  to  have  been 
committed  by  Maskclyne,  considering  the  accuracy  of  the 
observer  and  of  the  instruments,  and  the  large  number  of  ob- 
servations made.  Ilutton  suggests  the  repetition  of  the  ex- 
periment at  one  of  the  pyramids  in  Kgypt.  Some  years  later 
Peters  (80^)  made  a  calculation  of  the  attraction  of  the  Great 
Pyramid. 

For  brief  accounts  of  the  Schehallien  experiment  and  criti- 
cisms upon  it,  reference  should  be  made  to  Ilutton  (38  and  47, 
vol.  2,  pp.  G9-77),  von  Zach  (43,  44  and  V.)),  Muncke  (01,  vol. 
3,  pp.  944-70),  Schmidt  (64,  vol.  5J,  pp.  474-9),  Meiiabrea  (71), 

55 


\'\ 


\ 


ill 


fii 


i 


U' 


I'M 


i: 


l:i 


m 


mi 


!i 


I 


MKMOIliS    UN    THE    LAWS    OF    GRAVITATION 

Schell  (135),  Todliiinter  (140,  vol.  1,  pp.  459-60).  Ztinotti- 
Hiaiico  (I4Hi,  pt.  2,  pp.  125-35)  hikI   Fiesdorf  (ISfl^,  pp.  5-7). 

dipt.  J(u;ob  lias  remarked  (118  and  15il)tliat  by  this  method 
we  may  measure  the  attraction  of  the  mass  of  the  mountain 
above  the  surface,  yet  we  do  not  know  how  much  ouglit  to  be 
added  or  subtracted  due  to  that  delow  it. 

Von  Zach  makes  mention  of  several  early  astronomers  who 
assign  anomalies  in  theirgeodetic  measurements  to  the  influence 
of  mountains  on  the  plumb-lines  of  their  instruments;  the 
reader  is  referred  to  von  Zach,  Humboldt  (8!:i,  vol.  1,  notes, 
pp.  45-7)  and  Helmert  (148,  vol.  2,  chap.  4),  and  to  the  ac- 
count in  this  volume  (p.  123)  of  the  work  of  James  and 
Clarke.  Von  Zach  himself  niude  a  very  careful  determination 
in  1810,  after  the  method  used  by  Bouguer,  of  the  attraction 
of  mount  Mimet,  near  Murseilh^s.  He  found  a  deflection  of 
the  plumb-line  amounting  to  2".  lie  did  not  calculate  the 
density  of  the  earth.  His  observations  were  published  in  book 
form  in  1814  (41)). 

For  this  work  Maskelyne  was  presented  by  the  Iloyal  Society 
with  the  Copley  medal.  At  the  presentation  the  President,  Sir 
John  Pringle,  delivered  an  address  (35)  on  the  attraction  of 
gravitation,  giving  a  critical  account  of  the  state  of  the  subject 
before  the  time  of  Newton,  as  woU  as  of  hi  later  developments. 

56 


'  IW 


')     I.     ■:■ 


.4-: 


m 


EXPERIMENTS    TO    DETERMINE    THE 
DENSITY  OF  THE  EARTH 


BY 


HENRY   OAVENDISir,   Esq.,    F.R.S.   and  A.S. 

Read  June  21,  1798 


(From  t/w  PhilosophiMl  Tmnsaetiom  of  the  Ronnl  S^meU/  of  London  for  the 

year  1798,  Part  II.,  pp.  469-536) 


57 


(«f  t  ' 


.'*? 


.'^] 


If 


i    » 


CONTENTS 


PACK 


Introduction 59 

Description  of  the  appanituH 61 

Method  of  observing  the  deflection 64 

"    time  of  vibration 64 

Effect  of  the  resistance  of  the  air 65 

Account  of  the  experiments 67 

Testing  for  magnetic  effects 68 

Testing  tfie  elastic  properties  of  the  wire 72 

Further  tests  for  magnetic  effects 75 

Testing  the  effect  of  variation  of  temperature  about  the  box 76 

Final  observations 80 

On  the  tfieoi'g  of  the  erperiment 88 

Corrections  to  be  made  in  the  tlieory  as  first  given 91 

Effect  of  the  variable  position  of  the  arm  on  the  equations 97 

When  and  how  to  apply  the  corrections 98 

Table  of  results 99 

Conclusion 99 

Appendix  :  to  find  the  attraction  of  the  mahogany  case  on  the  balls 102 

58 


u 


EXPERIMENTS   TO    DETERMINE    THE 
DENSITY  OF  THE   EARTH 


BY 


HENRY    CAV^ENDISII,    Esq.,    F.H.S.    and    A.S. 


!:'■! 


i.  3 


Many  years  ago,  the  late  Rev.  John  Micliell,  of  this  society, 
nontrived  a  metliod  of  determiiiinf:^  tiie  density  of  the  eai  i,h,  hy 
rendering  sensible  the  attraction  of  small  (|uantities  of  matter  ; 
but,  as  he  was  engage<l  in  other  pursuits,  he  did  not  (iompiete 
the  apparatus  till  a  short  time  before  his  death,  aiul  did  .not 
live  to  make  any  experiments  with  it.  After  his  death,  the 
apparatus  came  to  the  Rev.  Francis  John  Hyde  Wollaston, 
Jacksonian  Professor  at  Cambridge,  Avho,  not  having  conven- 
iences for  making  experiments  with  it,  in  the  manner  he  could 
wish,  was  so  good  as  to  give  it  to  me. 

The  apparatus  is  very  simple  ;  it  consists  of  a  wooden  arm,  (j 
feet  long,  made  so  as  to  unite  great  strength  with  little  weight. 
This  arm  is  suspended  in  an  horizont'  1  jmsition,  by  a  slender 
wire  40  inches  long,  and  to  each  extremity  is  hung  a  leaden 
ball,  about  'I  inches  in  diameter  ;  and  the  whole  is  inclosed  in 
a  narrow  wooden  case,  to  defend  it  from  the  wind. 

As  no  more  force  is  required  to  make  this  arm  turn  round  on 
it  centre,  than  what  is  necessary  to  twist  the  suspending  wire, 
it  i-^  plain,  that  if  the  wire  is  sufficiently  slender,  the  most  min- 
ute force,  such  as  the  attraction  of  a  leaden  weight  a  few  inches 
in  diameter,  will  be  sufficient  to  draw  the  arm  sensibly  aside. 
The  weights  which  Mr.  Michell  intended  to  use  were  8  inches 
diameter.  One  of  these  was  to  be  placed  on  one  side  the  case, 
opposite  to  one  of  the  balls,  and  as  near  it  as  could  conveniently 
be  done,  and  the  other  on  the  other  side,  opposite  to  the  other 

59 


''Mi 


If 


1 1 


iVIKMOIUS    ON 


itiil 


pit  J 


ball,  so  tliJit  tlio  attraction  of  both  tlieso  weiglits  would  con- 
spire in  (Iniwiiig  the  arm  aside  ;  and,  when  its  position,  as  af- 
fected by  these  weights,  was  ascertained,  the  weights  were  to 
bo  removed  to  the  other  side  of  the  case,  so  as  to  draw  the  arm 
the  contrary  way,  and  the  position  of  the  arm  was  to  be  again 
determined  ;  and,  copseqiiently,  half  the  difference  of  these  po- 
sitions would  shew  how  much  the  arm  was  drawn  aside  by  the 
attraction  of  the  weights. 

In  order  Lo  determine  from  hence  the  density  of  tlie  earth, 
it  is  necessary  to  ascertain  what  force  is  required  to  draw  the 
arm  aside  through  a  given  s})ace.  This  Mr.  Michell  intended  to 
do,  by  putting  the  arm  in  motion,  and  observing  the  time  of  its 
vibrations,  from  which  it  may  easily  be  computed.* 

Mr.  Michell  had  prepared  two  wooden  stands,  on  which  the 
leaden  weights  were  to  be  su})ported,  and  j)ushed  forwards,  till 
they  came  almost  in  contact  with  the  case;  but  he  seems  to 
have  intended  to  move  them  by  hand. 

As  the  force  with  which  the  balls  are  attracted  by  these 
weights  is  excessively  minute,  not  more  than  6„, 00*0,000  ^^ 
their  weight,  it  is  plain,  that  a  very  minute  disturbing  force 
will  be  sufficient  to  destroy  the  success  of  the  experiment;  and, 
from  the  following  experiments  it  will  appear,  that  the  disturb- 
ing force  most  ditticult  to  guard  against,  is  that  arising  from 
the  variations  of  heat  and  cold  ;  for,  if  one  side  of  the  case  is 
warmer  than  the  other,  the  air  in  contact  with  it  will  be  rare- 
fied, and,  in  consequence,  will  ascend,  while  that  on  the  other 
side  will  descend,  and  produce  a  current  wliich  will  draw  the 
arm  sensibly  aside. f 

*  Mr.  Coulomb  lias,  in  a  variety  of  cases,  used  a  contrivance  of  this  kind 
for  trying  small  attractions  ;  but  Mr.  Micliell  informed  me  of  liis  intention 
of  making  lliis  experiment,  and  of  tbe  meiiiod  lie  intended  to  use,  before 
tbe  publication  of  any  of  Mr.  Coiilomb'.s  experiments. 

f  M.  Cassini,  in  observing  tbe  vnriation  compass  placed  by  him  in  tbe 
observatory  (wbieli  was  constructed  so  as  M^iivke  very  minute  changes  of 
position  visilde.  nnd  in  wliicli  tlie  needle  was  suspended  by  a  silk  thread), 
found  that  standing  near  tbe  box.  in  order  to  observe,  drew  tbe  needle 
sensibly  aside  ;  which  I  have  no  doubt  was  caused  by  this  current  of  air 
It  must  be  observed,  that  his  compass  box  was  of  metal,  which  transmits 
heat  faster  than  wood,  and  also  was  many  inches  deep  ;  both  which  causes 
served  to  increase  the  current  of  air.  To  diminish  the  effect  of  this  cur- 
rent, it  is  by  all  means  advisable  to  make  the  box,  in  which  the  needle 
plays,  not  much  deeper  than  is  necessary  to  prevent  the  needle  from  strik- 
ing against  the  lop  and  bottom. 

60 


T UK  I. A \v s  o I'  (1  li A \'  1  r A  r i o n 


As  I  was  convinced  of  the  necessity  of  jruunling  aj^ainst  this 
source  of  error,  1  resolved  to  place  the  a{)paratii8  in  a  room 
wiiicli  should  remain  constantly  slnit,  aiul  to  observe  the  mo- 
tion of  the  arm  from  without,  by  means  of  a  telescope  ;  and  to 
suspend  the  leaden  weij^fhts  in  such  manner,  that  I  could  move 
them  without  entering  into  the  room.  This  difference  in  the 
manner  of  observing,  rendered  it  necessary  to  make  some  al- 
teration in  Mr.  Michell's  apparatus  ;  and,  as  there  were  some 
I)arts  of  it  which  I  thought  not  so  convenient  as  could  bo 
wislied,  I  chose  to  make  the  greatest  part  of  it  afresh. 

Fig.  1  is  a  longitudinal  vertical  section  through  the  instru- 
ment, and  the  building  in  which  it  is  phuied.  AI5(Ji)I)CB- 
AEFFE  is  the  case  ;  .^•  and  x  are  the  two  balls,  whiidi  are  sus- 
pended by  the  wires  hx  from  the  arm  yhnih,  which  is  itself 
suspended  by  the  slender  wire  (jl.  This  arm  consists  of  a 
slen<ler  deal  rod  hinh,  strengthcniMl  by  a  silver  wire  hijh  ;  by 
wiiich  means  it  is  made  strong  enough  to  support  the  balls, 
though  very  light.* 

The  case  is  supported,  and  set  horizontal,  by  four  screws, 
resting  on  posts  fixed  firmly  into  the  ground  ;  two  of  them  are 
represented  in  the  figure,  by  S  and  S;  the  two  others  are  not 
represented,  to  avoid  confusion.  (J(r  and  (id  are  the  eiul 
walls  of  the  building.  W  and  W  are  the  leaden  weights ; 
which  are  suspended  by  the  copper  rods  IJrPrR,  and  the 
wooden  bar  rr,  from  the  centre  pin  Vp.  This  pin  passes 
through  a  hole  in  the  beam  II II,  perpendicularly  over  the  cen- 
tre of  the  instrument,  and  turns  round  in  it,  being  prevented 
from  falling  by  the  plate  p.  MM  is  a  pulley,  fastened  to  this 
pin;  and  Mm,  a  cord  wound  round  the  pulley,  and  passing 
through  the  end  w.all ;  by  which  the  observer  may  turn  it 
round,  and  thereby  move  the  weights  from  one  situation  to  the 
other. 

Fig.  3  is  a  plan  of  the  instrument.  AAAA  is  the  case.  SSSS, 
the  four  screws  for  supporting  it.  hh,  the  arm  and  balls.  W 
and  W,  the  weights.     MM,  the  pulley  for  moving  them.     When 


*Mr.  Michell's  rod  was  entirely  of  wood,  and  was  much  stronger  and 
stiffer  than  this,  thougli  not  much  heavier;  but,  as  it  had  warped  when  it 
came  to  me,  I  chose  to  make  another,  and  preferred  this  form,  partly  as 
being  easier  to  construct  and  meeting  with  less  resistance  from  the  air, 
and  partly  because,  from  its  being  of  a  less  complicated  form,  I  could  more 
easily  compute  how  much  it  was  attracted  by  the  weights. 

61 


'U 


MKiNJUlliS    ON 


lit 


rilK    IwWVS    OF    (}UA  VITATION 


tlio  weights  are  in  this  position,  l)()th  conspire  in  drawing  the 
arm  in  the  direetion  h\\  ;  hut,  when  they  are  removed  to  the 
situation  tn  and  //',  represented  hy  the  dotted  lines,  hoth  con- 
spire in  drawing  the  arm  in  the  contrary  direction  /nr.  These 
weights  are  prevented  from  striking  the  instrument,  in'  pieces 
of  wood,  wiiicii  stop  tiiem  as  soon  as  they  come  witiiin  I  of  an 
inch  of  the  case.     Tiie  pieces  of  wood  are  fastened  t)  the  wall 


,' -^^M 


Fig-  2 


of  the  building;  and  I  find,  that  the  weights  may  strike  against 
them  with  considerable  force,  without  sensibly  shaking  the  in- 
strument. 

In  order  to  determine  the  situation  of  the  arm,  sli})s  of  ivory 
are  placed  within  the  case,  as  near  to  each  end  of  the  arm  as 
can  be  done  without  danger  of  touching  it,  and  are  divided  to 
20ths  of  an  inch.  Another  small  slip  of  ivory  is  placed  at  each 
and  of  the  arm,  serving  as  a  vernier,  and  subdividing  these 
divisions  into  5  parts  ;  so  that  the  position  of  the  arm  may  be 
observed  with  ease  to  lOOths  of  an  inch,  and  may  be  estimated 
to  less.  These  divisions  are  viewed,  by  means  of  the  short 
telescopes  T  and  T  (Fig.  1),  tiirougli  slits  cut  in  the  end  of  the 
case,  and  stopped  with  glass;  they  are  enlightened  by  the  lamps 
L  and  L,  with  convex  glasses,  placed  so  as  to  throw  the  light 
on  the  divisions  ;  no  other  light  being  admitted  into  the  room. 

The  divisions  on  the  slips  of  ivory  run  in  the  direction  Ww 
(Fig.  2),  so  that,  when  the  weights  are  placed  in  the  positions 
wand  w,  represented  by  the  dotted  circles,  the  arm  is  drawn 
aside,  in  such  direction  as  to  make  the  index  point  to  a  higher 
number  on  the  slips  of  ivory  ;  for  which  reason,  I  call  this  the 
positive  position  of  the  weights. 

FK  (Fig.  1)  is  a  wooden  rod,  which,  by  means  of  an  endless 
screw,  turns  round  the  support  to  which  the  wire  yl  is  fastened, 

63 


■M 


>* 


_im.-jjv.w:^-i 


'U 


i; 


MKMOIKS    UN 


tl 


and  tlier('})y  oniiblos  tlie  obsorvor  to  turn  round  tho  wire,  till 
the  arm  settloH  in  tlio  iniddht  <tf  tli(>  cnao,  without  danger  of 
touching?  oitlier  side.  Tiio  win?  ///  is  fastened  to  its  support  at 
top,  and  to  the  centre  of  tlie  arm  at  bottom,  by  brass  clips,  in 
whicli  it  is  pin(died  by  sorows. 

In  these  two  figures,  the  difTorent  ])arta  are  drawn  nearly  in 
the  proper  proportion  to  eaeli  otiier,  and  on  a  scale  of  one  to 
tiiirteen. 

Before  I  proceed  to  the  account  of  the  experiments,  it  will 
be  pro))er  to  say  something  of  the  manner  of  observing.  Sup- 
pose the  arm  to  be  at  rest,  and  its  position  to  be  observed,  let 
the  weights  be  then  moved,  tlie  arm  will  not  only  be  drawn 
aside  thereby,  but  it  will  be  made  to  vibrate,  and  its  vibrations 
will  continue  a  great  while  ;  so  that,  in  order  to  determine  how 
much  the  arm  is  drawn  aside,  it  is  necessary  to  observe  the  ex- 
trenie  points  of  the  vibrations,  and  from  thence  to  determine 
the  point  which  it  would  rest  at  if  its  motion  was  destroyed,  or 
the  point  of  rest,  as  I  shall  call  it.  To  do  this,  I  observe  three 
successive  extreme  points  of  a  vibration,  and  take  the  mean 
between  the  first  ami  third  of  these  points,  as  the  extreme 
point  of  vibration  in  one  directioii,  and  then  assume  the  mean 
between  this  and  the  second  extreme,  as  the  point  of  rest ;  for, 
as  the  vibrations  are  continually  diminishing,  it  is  evident,  that 
the  mean  between  two  extreme  points  will  not  give  the  true 
point  of  rest. 

It  may  be  thought  more  exact,  to  observe  many  extreme 
points  of  vibration,  so  as  to  find  the  point  of  rest  by  different 
sets  of  three  extremes,  and  to  take  the  mean  result ;  but  it 
must  be  observed,  that  notwithstanding  the  pains  taken  to  pre- 
vent any  disturbing  force,  the  arm  will  seldom  remain  perfect- 
ly at  rest  for  an  hour  together;  for  which  reason,  it  is  best  to 
determine  the  point  of  rest,  from  observations  made  as  soon 
after  the  motion  of  the  weights  as  possible. 

The  next  thing  to  be  determined  is  the  time  of  vibration, 
which  I  find  in  this  manner:  I  observe  the  two  extreme  points 
of  a  vibration,  and  also  the  times  at  which  the  arm  arrives  at  two 
given  divisions  between  these  extremes,  taking  care,  as  well  as  I 
can  guess,  that  these  divisions  shall  be  on  different  sides  of  the 
middle  point  and  not  very  far  from  it.  I  then  compute  the 
middle  point  of  the  vibration,  and,  by  proportion,  find  the  time 
at  which  the  arm  conies  to  this  middle  point.     I  then,  after  a 

64 


TIIK    LAWS    OK    (JliA  V  ITATloN 


number  of  vibrutioiis,  ropoat  this  oj)omtion,  and  divide  the  in- 
terval of  time,  between  the  cominjjf  of  the  arm  to  these  two 
middle  points,  by  tlio  number  of  v'brations,  whieh  jjives  the 
time  of  one  vibration.  The  following  example  will  explain 
wiiat  is  here  said  more  clearly : 


Fxtreiiu! 
poiiitH 

DiviHioua 

Tinio 

I'oiht  of 

Timi>  (irmltlillc 
(if  vihrttlKiK 

27.2 

25 
24 

10"  2:r   4  ) 

57    \ 

— 

10''  2:j  23 ' 

22  1 

— 

24  <l 

27. 

— 

— 

24  7 

22.6 

— 

— 

24  75 

2tJ.H 

— 

— 

24.8 

2:}. 

— 

— 

24.85 

26.6 

— 

— 

24.9 

25 
24 

11      5   22    ) 

6   48 

— 

11      5   22 

23.4 

The  first  column  contains  the  extreme  points  of  the  vibra- 
tions. The  second,  the  intermediate  divisions.  The  third, 
the  time  at  whinh  the  arm  came  to  these  divisions ;  and  the 
fourth,  the  point  of  rest,  which  is  thus  found  :  the  mean  be- 
tween the  first  and  third  extreme  points  is  ^7.1,  and  the  mean 
between  this  and  the  second  extreme  point  is  ;^4.(),  which  is 
the  point  of  rest,  as  found  by  the  three  first  extremes.  In  like 
manner,  the  point  of  rest  found  by  the  second,  third,  and 
fourth  extremes,  is  24.7,  and  so  on.  The  fifth  column  is  the 
time  at  which  the  arm  came  to  the  middle  point  of  the  vibra- 
tion, which  is  thus  found-:  the  mean  between  27.2  and  22.1  is 
24.05,  and  is  the  middle  point  of  the  first  vibration  ;  and,  as 
the  arm  came  to  25  at  10''  23'  4",  and  to  24  at  10''  23'  57",  we 
find,  by  proportion,  that  it  came  to  24. G5  at  10''  23'  23".  In 
like  manner,  the  arm  came  to  the  middle  of  the  seventh  vibra- 
tion at  11''  5'  22";  and,  therefore,  six  vibrations  were  performed 
in  41'  59",  or  one  vibration  in  7'  0". 

To  judge  of  the  propriety  of  this  method,  we  must  consider 
in  what  manner  the  vibration  is  affected  by  the  resistance  of  the 
air,  and  by  the  motion  of  the  point  of  rest. 

Let  the  arm,  during  the  first  vibration,  move  from  D  to  B 
(Fig.  3),  and,  during  the  second,  from  B  to  ^/ ;  Bd  being  less 
than  DB,  on  account  of  the  resistance.  Bisect  DB  in  M,  and 
B  65 


f 
It 


H 


'{\ 


M KM OIKS     UN 

IW  in  m,  luul  bisect  M///  in  n,  and  let  .r  ho  nny  ])oint  in  the 
vihnition ;  tlion,  if  the  rcsistiiiico  is  pioportionul  to  the  HCjuaro 
of  the  veU)(!ity,  the  whole  time  of  a  vihnition  iu  very  little  al- 
tered ;  hut,  if  T  '^}  taken  to  the  time  of  one  vihration,  aa  tiio 
diameter  of  a  circle  to   its  Hemi-circuimference,  the  time   of 

moving;  from  \\  to  n  ex(!ced8  \  a  vihration,  hy        ..        nearly  ; 

and  the  time  of  moving  from  B  to  m  falls  short  of  I  a  vihration, 


M 


-I — f- 


fi 


n    m 


*m 


by  as  much ;  and  the  time  of  moving  from  H  to  .r,  in  the  sec- 
ond vihnition,  exceeds  that  of  moving  from  ;/•  to  H.  in  the  first, 

TxD^/xHr' 
by , "  supposing  \)d  to  be  bisected  in  h  ;  so  that, 

if  a  mean    is   taken,  between   the   time  of  the  first  arrival  of 
the  arm  at  x  and  its  returning  hack  to  the  same  point,  this 
moan  will  be  earlier*  than  the  true  time  of  its  coming  to  B,  by 
TxPr/xBa;" 

The  effect  of  motion  in  the  point  of  rest  is,  that  when  the 
arm  is  moving  in  the  same  direction  as  the  point  of  rest,  the 
time  of  moving  from  one  extreme  point  of  vibration  to  the 
other  is  increased,  and  it  is  diminished  when  they  are  moving 
in  contrary  directions  ;  but,  if  the  point  of  rest  moves  uni- 
formly, the  time  of  moving  from  one  extreme  to  the  middle 
point  of  the  vibration,  will  be  equal  to  that  of  moving  from 
the  middle  point  to  the  other  extreme,  and,  moreover,  the  time 
of  two  successive  vibrations  will  be  very  little  altered  ;  and, 
therefore,  the  time  of  moving  from  the  middle  point  of  one 
vibration  to  the  middle  point  of  the  next,  will  also  be  very 
little  altered. 


*  [ThiH  word  should  be  "later"  as  is  observed  by  TodJiunter  (140.  vol.  2,  p. 
KJ;")).  For  an  elementary  discxission  of  this  kind  of  motion  see  Williamson  and 
Tarletoii's  "  Treatise  of  Dynamics"  e.v.  13,  ii?  117  Poisson  (65,  vol.  1,  pp. 
353-36i)  and  Menabrea  (71)  have  given  very  elaftorate  analyses  of  the  problem. 
Cornu  and  Bailie  (137,  141,  142,  143.  and  157)  jn-oved  in  1878  tfiat  the  re- 
sistance in  tJi€  case  under  consideration  is  proportional  to  tlie  first  power  of 
tJis  velocity.^ 

66 


TIIK    LAWS    OK    (JKA  V  ITATIoN 


It  appotirs,  tliereforc,  that  on  account  of  the  resistance  of 
the  air,  the  time  at  whi(;h  the  arm  comes  to  the  middle  point 
of  the  vihration,  is  not  exactly  the  mean  hetween  th«'  times  of 
its  coming  to  thi;  extreme  points,  wliii'h  «'aiises  some  inaccur- 
acy in  my  method  of  lindin^  the  time  of  a  vihration.  It  must 
he  ohservcd,  however,  that  as  the  time  of  (Mjinin^'  to  the  miihlh- 
point  is  hoforo  tiie  middle  of  the  vihration,  hoth  in  tint  lirst 
and  last  vihration,  and  in  ^^'ueral  is  nearly  e(|ually  so,  the  error 
produced  from  this  cause  must  he  inconsiderahhs  and,  on  the 
whole,  I  see  no  method  of  finding  the  tinn^  of  a  vihration  whi(di 
is  liahle  to  less  ohjection. 

'I'he  time  of  a  vihration  tmiy  he  determined,  either  hy  previous 
trials,  or  it  may  he  done  at  each  experiment,  hy  ascertaining  the 
time  of  the  vihrations  which  the  arm  is  actually  put  into  by 
the  motion  of  the  weights  ;  hut  there  is  one  advantage  in  the 
latt(>r  method,  luimely,  that  if  there  should  bo  any  accidental 
attra(!tion,  such  as  electricity,  in  the  glass  })late8  through  which 
the  motion  of  the  arm  is  seen,  which  should  increase  the  forcse 
necessary  to  draw  the  arm  aside,  it  wouhl  also  diminish  tlie 
time  of  vibration  ;  and,  consequently,  the  error  in  the  result 
would  be  much  loss,  when  the  force  required  to  draw  the  ami 
aside  was  deduced  from  experiments  made  at  the  time,  than 
when  it  was  taken  from  previous  experiments. 


If 


a 


ACCOFNT   OF   TIIK   EXPERIMENTS 


re- 
of 


In  my  first  experiments,  the  wire  by  which  the  arm  was  sus- 
pended was  3()|  inches  long,  and  was  of  copper  silvered,  one 
foot  of  whicii  weighed  2^  grains  ;  its  stiffness  was  such  as  to 
make  the  arm  perform  a  vibration  in  about  15  minutes.  I  im- 
mediately found,  indeed,  that  it  was  not  stiff  enough,  as  tlie 
attraction  of  the  weights  drew  the  balls  so  much  aside,  as  to 
make  them  touch  the  sides  of  the  case  ;  I,  however,  chose  to 
make  some  experiments  with  it,  before  I  changed  it. 

In  this  trial,  the  rods  by  which  the  leaden  weights  were  sus- 
pended were  of  iron  ;  for,  as  I  had  taken  care  that  there  should 
be  nothing  magnetical  in  the  arm,  it  seemed  of  no  signification 
whether  the  rods  were  magnetical  or  not ;  but,  for  greater  se- 
curity, I  took  off  the  leaden  weights,  and  tried  what  effect  the 
rods  would  have  by  themselves.  Now  I  find,  by  computation, 
that  the  attraction  of  gravity  of  these  rods  on  the  balls,  is  to 

67 


MKMOlltS    ON 


Hi 


I'i! 


II 


tliiit  of  tlio  W(i|,'fjtH,  iHMirly  us  17  to  '<ir)00  ;  ho  that,  hh  the  ut- 
tni(;tioii  of  the  wri^lits  iippeiircMl,  l»y  tlio  fon';^'oiM;;  triiil,  to  he 
Hiiniciciit  to  ciniw  tin;  urrii  asido  l»y  about  15  tlivi.sioiiH,  tliu  at- 
trai^tioii  of  \\h'  rotl.H  jiloiic  kIioiiM  draw  it  asidi;  about  ^^  of  a 
division  ;  and,  tiKM'oforo,  tbo  iiiotioii  of  tbo  rotls  from  one  iiuar 
position  to  till!  otb(!r,  should  move  it  about  ^  of  a  division. 

'I'ho  result  of  tho  oxpcrinu-nt  was,  that  for  the  llrst  ir»  niin- 
ntus  after  tin;  rods  were  rutiiovod  from  om;  nuur  position  to  thu 
othor,  very  little  motion  was  produced  in  the  arm,  and  hardly 
mor(!  than  ou^ht  to  be  produc^ed  by  the  action  of  gravity  ;  but 
the  motion  tluMi  increased,  so  that,  in  about  a  (juarter  or  luilf 
an  hour  more,  it  was  found  to  have  moved  ^  or  [^  division,  in 
tlio  same  ilirection  that  it  ought  to  have  done  by  tho  action  of 
gravity.  On  returning  the  irons  ba(!k  to  their  former  position, 
tho  arm  moved  backward,  in  the  same  manner  that  it  before 
moved  forward. 

It  must  be  observed,  that  tho  motion  of  tho  arm,  in  tlioso  ex- 
periments, was  hardly  more  than  would  sometimes  take  place 
without  any  apparent  cause  ;  but  yet,  as  in  tlirec  experiments 
which  wore  made  with  those  rods,  tho  motion  was  constantly  of 
the  same  kind,  though  differing  in  quantity  from  ^  to  1^  divis- 
ion, tiiero  seems  great  reason  to  think  that  it  was  produced  by 
tlio  rods. 

As  this  effect  seemed  to  mo  to  be  owing  to  magnetism,  though 
it  was  not  such  as  I  should  have  expected  from  that  cause,  I 
changed  the  iron  rods  for  copper,  and  tried  them  as  before  ; 
tho  result  was,  that  there  still  sconiod  to  bo  some  effect  of  the 
same  kind,  but  more  irregular,  so  that  I  attributed  it  to  some 
accidental  cause,  and  therefore  hung  on  tho  leaden  weights,  and 
proceeded  with  tho  experiments. 

It  must  bo  observed,  that  tho  effect  which  seemed  to  be  pro- 
duced by  moving  tho  iron  rods  from  one  near  i)08ition  to  tho 
other,  was,  at  a  medium,  not  more  than  one  division  ;  whereas 
the  effect  produced  by  moving  tho  weight  from  the  midway  to 
tho  near  position,  was  about  15  divisions  ;  so  that,  if  I  had  con- 
tinued to  use  the  iron  rods,  the  error  in  tho  result  caused  there- 
by, could  hardly  have  exceeded  -^^  of  the  whole. 

68 


Til  K    LAWS    OF   <Jli.\  VITATION 


ExpKuixfKMT  I.     Aro.  5 

Wi  iifht»  in  Viiihnifi  iumtion 


KxlDMIIA 
|Nllll(M 


PIvImIoiih 

Timt 

ri.4 

11.5 
11.5 

li''  42     0 

55     0 

U)      5     0 

I'ollil  ol 

rcHi 


11.0 


Tlino  of  mill  of 
vlhrallou 


Diflvrenc* 


23.4 
27.0 
24.7 
27  :i 
25.1 


At  10''  5  ,  weiyhtH  mured  to  ponUive  jHmtion 


25.  H2 
2«l.()7 
20.1 


Atn 


18.2 

GO 

10.8 

7.7 


11 
12 

12 
11 

11 
12 

12 
11 


••  0',  ireif/htH  returned  Ixick  to  midwdy  jtosition 

0"     1    13 


0      0   4S    i 
1    30    S 


10  20  i 

17  20  S 

30  24  ) 

31  11  ) 

45  58  / 

47  4  )" 


12 
11.72 


10     9 
30   45 

45   58 


14  56 ' 

14  30 

15  13 


Motioei  on  moving  froni  midway  to  pos.  =  14.32 
'  pos.  to  midway  =  14.1 

Time  of  ono  vilnation  =  14'  55" 

It  must  be  observed,  tlmt  in  this  experiment,  the  attnictioii 
of  the  weiglits  drew  tlie  arm  from  11.5  to  )ir>.H,  so  that,  if  no 
contrivance  had  been  used  to  prevent  it,  the  momentum  ac- 
quired thereby  would  have  carried  it  to  near  40,  and  would, 
therefore,  liave  made  the  balls  to  strike  against  the  case.  To 
prevent  this,  after  the  arm  had  moved  near  l."»  divisions,  I  re- 
turned the  weights  to  the  midway  position,  and  let  them  re- 
main there,  till  the  arm  came  nearly  to  the  extent  of  its  vibra- 
tion, and  then  again  moved  them  to  the  positive  position, 
whereby  the  vibrations  were  so  much  diniinishiMl,  that  the  balls 
did  not  touch  the  sides;  and  it  was  this  which  ])revented  my 
observing  the  first  extremity  of  the  vibration.  A  like  method 
was  used,  when  the  weights  were  returned  to  the  midway  posi- 
tion, and  in  the  two  following  experiments. 

69 


fill  I 


11 


-!! 


rrv 


MKMOIRS    ON 

The  vibrations,  in  moving  the  weights  from  the  midway  to 
the  positive  position,  were  so  small,  that  it  was  thought  not 
worth  while  to  observe  the  time  of  the  vibration.  When  the 
weights  were  returned  to  the  midway  position,  I  determined 
the  time  of  the  arm's  coming  to  the  middle  pcint  of  each  vibra- 
tion, in  order  to  see  how  nesirlv  the  times  of  the  different 
vibrations  jigreed  together.  In  great  part  of  the  following  ex- 
])eriments,  1  contented  myself  with  observing  the  time  of  its 
coming  to  the  middle  point  of  only  the  first  and  last  vibration. 

EXFKUIMENT    II.       AUO.  G 
Weigh'"  in  midway  position 


Fxtrerne 
poiiitH 

DiviHiijiiH 
11 

Time 

I'oilil  ot 
rest 

Time  of  mid   ol 
vibration 

DifTeronce 

lOh     4'     0' 

11 

11       0 

11 

17     0 

11 

25     0 

11. 

Weights  moved 

to  positive  fMsition 

29.3 

24.1 

— 

— 

26.87 

30. 

.. 

— 

27.57 

r.2 

— 

— 

28.02 

V  -.7 

— 

— 

28.12 

2(5. !» 

— 

— 

28.05 

28.7 

— 

27.85 

27.1 

— 

27.82 

28.4 

Weights  returned 

''  to  midway  position 

6. 

12 

1      3    50    \ 

1"    4'    1" 

13 

4   34    f 

18.5 

— 

— 

12.37 

— 

>    14' 52' 

13 

18   29    } 

18    53 

12 

19    18    f 

6  5 

— 

— 

11.67 

— 

14  46 

11 
12 

33  48    \ 

34  51    S 

— 

33   39 

15.2 

— 

— 

11. 

— 

13  46 

13 
12 

45  8    ,) 

46  22    S 

— 

47   25 

7.1 

— 

— 

10.75 

— 

15  25 

11 

12 

3      3   48    / 
5   18    S 

— 

2      2   50 

13.6 

Motion  of  arm  on  movins?  weights  from  midway  to  po.s.  =  15  87 

pos.  to  midway  —  15.45 
Time  of  one  vibration  =  14'  42  ' 

70 


It-  '!■ 


T  UK    L  A  W  S    ( )  F    < ;  K  A  \'  I  T  A  1'  I  O  N 


125 


ExpKiiiMKNT   III.     Arc;.  7 
7'he  weifjhfn  ha'nf/  in  the  ponitiee  ixisitioii,  and  the  arm.  a  little  in  motion 


Kxin'iiie 

ni 

visions 

|MIIIIIS 

Hl.T) 

29 

— 

»1 

— _ 

29.1 

Timo 


I'oint  of 


Tiinn  of  mill,  of 
vibration 


9. 
20.5 

9.2 

17.4 

10.1 

15.6 
32. 


23.7 
31.8 
25.8 


31.1 


14 
15 

15 
14 

14 
15 

It 
13 

13 
14 


28 

27 


27 
28 


30.12 
30.02 


Weights  moved  to  midway  jjosition 

10"  34'  55 " 


DiflTerenco 


10"  34'  18"  ) 
35  6  f 


49  31  / 

50  27  f 

11   5  7  } 

6  18  <i 

18  46  / 

19  58  \ 

33  46  / 

35  26  S 


14.8 
14.07 
13.52 
13.3 


49  39 

11   4  17 

19  4 

33  31 


14'  44 ' 
14  38 
14  47 
14  37 


Weights  moved  lo  positive  position 

0      2   59 

47   40 


0      2   48    / 

3   56    S 

— 

27.8 

— 

28.27 

— 

28.62 

44   58    1 

46   50    S 

Motion  of  the  arm  on  moving  weights  from  pos.  to  mifi.=  15.22 

mid.  tf)  pcKS.  =  14.5 
Time  of  one  vibration,  wlicn  in  mid.  position  —  14  39" 

pos.  position  =  14'  54  " 


Tliese  experiments  are  sufficient  to  shew,  tiiat  tlie  attraction 
of  tlie  weights  on  the  balls  is  very  sensible,  and  are  also  suf- 
ficiently regular  to  determine  the  quantity  of  this  attraction 
pretty  nearly,  as  the  extreme  results  do  not  differ  from  each 
other  by  more  than  ^  part.  But  there  is  a  circumstance  in 
them,  the  reason  of  which  does  not  readily  appear,  namely, 
that  the  effect  of  the  attraction  seems  to  increase,  for  half  an 

71 


t^ 


I '. 


i.i'i 


MEMOIRS    ON 


1 


1^       M-'fi    ! 


honr,  or  an  hour,  after  the  motion  of  the  weights ;  as  it  may 
be  observed,  that  in  all  three  experiments,  the  mean  position 
kept  increasing  for  that  time,  after  moving  the  weigiits  to  the 
positive  position  ;  and  kept  decreasing,  after  moving  them 
from  the  positive  to  tlie  midway  position. 

Tiie  first  cause  wliicii  occurred  to  me  was,  that  possibly  there 
might  be  a  want  of  elasticity,  either  in  the  suspending  wire,  or 
something  it  was  fastened  to,  which  might  make  it  yield  more 
to  a  given  pressure,  after  a  long  continuance  of  that  pressure, 
than  it  did  at  first. 

To  put  this  to  the  trial,  I  moved  the  index  so  much,  that  the 
arm,  if  not  prevented  by  the  sides  of  the  case,  would  have  stood 
at  about  50  ('"visions,  so  that,  as  it  could  not  move  farther  than 
to  35  divisiouo,  it  was  kept  in  a  position  15  divisions  distant 
from  that  which  it  would  naturally  have  assumed  from  the 
stiffness  of  the  wire  ;  or,  in  other  words,  the  wire  was  twiste<l 
15  divisions.  After  having  remained  two  or  three  hours  in  this 
position,  the  index  was  moved  back,  so  as  to  leave  the  arm  at 
liberty  to  assume  its  natural  position. 

It  must  be  observed,  that  if  a  wire  is  twisted  only  a  little 
more  than  its  elasticity  admits  of,  then,  instead  of  setting,  as 
it  is  called,  or  acquiring  a  permanent  twist  all  at  once,  it  sets 
gradually,  and,  when  it  is  left  at  liberty,  it  gradually  loses  part 
of  that  set  which  it  acquired  ;  so  that  if,  in  this  experiment, 
the  wire,  by  having  been  kept  twisted  for  two  or  three  hours, 
lia<l  gradually  yielded  to  this  pressure,  or  had  begun  to  set,  it 
would  gradually  restore  itself,  when  left  at  liberty,  aiul  the 
point  of  rest  would  gradually  move  backwards  ;  but,  though 
the  experiment  was  twice  repeated,  I  could  not  ))crceive  any 
such  efff'ct. 

The  arm  was  next  suspended  by  a  stiffer  wire. 


1  ^ 


Experiment  IV.     Aug.  12 
Weights  in  midway  position 


Kxtrcnie 
poiDts 


Divisions 


21.6 
21.5 
21.5 


Time 

9''  30'    0" 
52     0 
10    13    0 


I'oint  of 
rest 


21.5 


Time  of  mid.  of 
vibration 


Difference 


7» 


i«f    i 


' 


TIIK    LAWS    OK    (IIIAVITATION 


Weif/htH  morcdfroin  vtidiray  to  poKitive  ponidon 


it, 

!VS, 

it 


ce 


27.2 

22.1 

— 

— 

24.6 

27. 

— 

— 

24.(57 

22  6 

— 

— 

24.75 

26  8 

— 

— 

24  8 

28  () 

— 

— 

24.85 

26.6 

— 

— 

24  9 

23.4 

Wt'if/hts  mo  red  to  ntf/dlire  ]if)iiitiofi 

15. 

17 
19 

19  25    } 

20  41     f 

— 

10"  20'  31  ' 

22.4 

— 

— 

18.72 

— 

7'    0' 

20 
19 

26  45    } 

27  22    f 

— 

27    81 

15.1 

— 

— 

18.52 

— 

6  57 

19 
20 

35     1     / 

48    f 

— 

34   28 

21.5 

~~~ 

— 

18.85 

— 

7  23 

20 
19 

40  23    } 

41  18    <i 

— 

41    51 

15.8 

— 

— 

18.22 

— 

6  48 

18 
19 

48  36    } 

49  24    f 

— 

48   89 

20.8 

— 

— 

18.1 

— 

6  58 

19 

18 

54  45    / 

55  45    S 

— 

55   37 

15.5 

Wcir/htH  moved 

to  positive  position 

31.3 

25 
28 

11    10  25    } 
11     3    f 

— 

11    10   40 

17.1 

— 

— 

24.02 

— 

7    3 

22 
28 

17     6    1 
26    S 

— 

17   48 

30.6 

— 

— 

24.17 

— 

7    1 

25 
28 

24  83    ) 

25  17    f 

— 

24   44 

18.4 

— 

— 

24  32 

— 

7    5 

23 
25 

31  21     } 

32  9    S 

— 

81    49 

29.9 

— 

— 

244 

— 

6  59 

25 
28 

88   89    ) 
39  31     \ 

— 

38   48 

19.4 

— 

— 

24  5 

— 

7     6 

23 
25 

45  16    / 

46  12    \ 

— 

45   54 

29.3 

Moiion  of  arm  on  moving  weights  from  midway  to  pos.  —  3.1 

pos.  to  nei?.  =  6.18 
iieg.  lo  pos.  r- 5.92 

Time  of  one  vibration  in  neg.  position  —7'  l" 

pos.  position 


;     it 


f 


Iff 


'^TT'rrfV'  |  I  IPP  i^V    ''"'•»!,  ^.' I 


: 


MEMOIRS    ON 

Experimf<:nt  V.     Aru.  20 

7'/<«  weights  being  in  the  pfmtive  poHitinn,  the,  urm  wan  made  to  vibrate,  by 

intiiHiig  the  iiitirx 


Kxtromo 
points 

DiviHions 

Time 

h.int  or 

roHt 

TiMio  of  mid.  of 
vibration 

DifTerence 

29.6 

21.1 

— 

— 

25.2 

29. 

— 

— 

25.17 

21.6 

Weightu  iiiored  to  negative  jwsition 

22.6 

20 
19 

10"  22'    47  "  } 
23    30    S 

— 

10"  23'    11 " 

16.3 

— 

— 

19.27 

21.9 

— 

— 

19.15 

16.5 

— 

— 

19.1 

21.5 

— 

— 

19.07 

16.8 

— 

— 

19.07 

21.2 

— 

— 

19.07 

17.1 

— 

— 

19.05 

20.8 

— 

— 

19.02 

17.4 

— 

— 

19.05 

20.6 

— 

— 

19.02 

20 
19 

11     32     16    \ 
33    58    S 

~ 

11     33    53 

17.5 

— 

— 

18.97 

— 

7'    13" 

19 

41     16    / 

41       6 

20 

43      0    f 

20.3 

Weight f<  moved 

to  jioHitire  jWHition 

20.2 

24 
26 

49  10    / 

50  19    ( 

— 

49    37 

29.4 

— 

— 

24.95 

7    7 

26 
25 

56     15    / 
47    S 

— 

56    44 

20.8 

_.. 

24  92 

28.7 

— 

— 

24.87 

21.3 

— 

— 

24.85 

28.1 

— 

— 

24.75 

21.5 

— 

— 

24.67 

27.6 

— 

— 

24.67 

22. 

— 

— 

24.7 

24 

0    45    48    } 
46    43    \ 

— 

0    46    21 

27.2 



— 

24.7 

— 

7    1 

25 
24 

53  11    ) 

54  9    J' 

— 

53    22 

22.4 

Motion  of  arm  on  moving  weights  from  pos.  to  neg.=  5.9 

neg.  to  po8.  =  5.98 
Time  of  one  vibration,  when  weights  are  in  neg.  position  =  7' 5" 
pos.  position  —  T  5" 

74 


TIIK    LAWS    OF    (iUAVlTATlON 


In  the  fourtli  oxperiment,  the  effect  of  the  weights  seemed 
to  increase  on  stjinding,  in  all  tliree  motions  of  the  weiglits, 
conformably  to  wliat  was  observed  with  the  former  wire  ;  but 
in  tlie  last  experinjent  tiie  case  was  dilfereTit ;  for  though,  on 
moving  the  weiglits  from  positive  to  negative,  the  effect  seemed 
to  increase  on  standing,  yet,  on  moving  them  from  negative  to 
positive,  it  diminislied. 

My  next  trials  were,  to  see  whether  this  effect  was  owing  to 
magnetism.  Now,  as  it  happened,  the  case  in  which  the  arm 
was  inclosed,  was  placed  nearly  parallel  to  the  magnetic;  east, 
and  west,  and  therefore,  if  tiiere  was  anything  magnetic  in  the 
balls  and  weights,  the  balls  would  acquire  polarity  from  the 
earth  ;  and  the  weights  also,  after  having  remained  some  time, 
either  in  the  positive  or  negative  position,  would  acquire  po- 
larity in  the  same  direction,  and  would  attract  the  balls;  but, 
when  the  weights  were  moved  to  the  contrary  position,  that 
pole  which  before  pointed  to  the  north,  would  point  to  the 
sonth,  and  would  repel  the  ball  it  was  approached  to  ;  but  yet, 
as  repelling  one  hall  towards  the  south  has  the  same  effect  on 
the  arm  as  attracting  the  other  towards  the  north,  this  would 
have  no  effect  on  the  position  of  the  arm.  After  some  time, 
however,  the  poles  of  the  weight  would  be  reversed,  and  would 
begin  to  attract  the  balls,  and  would  therefore  produce  the 
same  kind  of  effect  as  was  actually  observed. 

To  try  whether  this  was  the  case,  I  dettiched  the  weights 
from  the  npper  })art  of  the  copper  rods  by  which  they  were 
suspended,  but  still  retained  the  lower  joint,  namely,  that 
which  passed  through  them  ;  I  then  fixed  them  in  their  posi- 
tive position,  m  such  manner,  that  they  could  turn  round  on 
this  joint,  as  a  vertical  axis.  I  also  made  an  apparatus  by 
which  I  could  turn  them  half  way  round,  on  these  vertical  axes, 
without  opening  the  door  of  the  room. 

Having  suffered  the  apparatus  to  remain  in  this  manner  for  a 
day,  I  next  morning  observed  the  arm,  and,  having  found  it  to 
be  stationary,  turned  the  weights  half  way  round  on  their  axes, 
but  could  not  perceive  any  motion  in  the  arm.  Having  suf- 
fered the  weights  to  remain  in  this  position  for  about  an  hour, 
I  turned  them  back  into  their  former  position,  but  without  its 
having  any  effect  on  the  arm.  This  experiment  was  repeated 
on  two  other  days,  with  the  same  result. 

We  may  be  sure,  therefore,  that  the  effect  in  question  could 

75 


MEMOIRS    ON 


.    1'! 


not  be  produced  by  magnetism  in  tbe  woiglits  ;  for,  if  it  was, 
turning  them  half  round  on  tlieir  jt.xe«,  would  immediately  have 
changed  tlieir  magJietic  attraction  into  repulsion,  and  have 
produced  a  motion  in  the  arm. 

As  a  further  proof  of  this,  I  took  off  th(  leaden  weights,  and 
in  tlieir  room  placed  two  10-inch  magnets  ,  the  apparatus  for 
turning  them  round  being  left  as  it  was,  and  the  magnets  being 
placed  horizontal,  and  pointing  to  the  balls,  and  with  their 
north  poles  turned  to  the  north  ;  but  I  could  not  find  that 
any  alteration  was  produced  in  the  place  of  the  arm,  by  turn- 
ing them  half  round  ;  which  not  only  confirms  the  deduction 
drawn  from  the  former  experiment,  but  also  seems  to  shew,  that 
in  the  experiments  with  the  iron  rods,  the  effect  produced  could 
not  be  owing  to  magnetism. 

The  next  thing  which  suggested  itself  to  me  was,  that  pos- 
sibly the  effect  might  be  owing  to  a  difference  of  tempera- 
ture between  the  weights  and  the  case  ;  for  it  is  evident, 
that  if  the  weights  were  mnch  warmer  than  the  case,  they 
would  warm  that  side  which  was  next  to  them,  and  prodnce  a 
current  of  air,  which  would  make  the  balls  approach  nearer  to 
the  weights.  Though  I  thought  it  not  likely  that  there  should 
be  sufficient  difference,  between  the  heat  of  the  weights  and 
case,  to  have  any  sensible  effect,  and  though  it  seemed  im- 
probable that,  in  all  the  foregoing  experiments,  the  weights 
should  happen  to  be  warmer  than  the  case,  1  resolved  to  ex- 
amine into  it,  and  for  this  purpose  removed  the  apparatus  nsed 
in  the  last  experiments,  and  supported  the  weights  by  the  cop- 
per rods,  as  before  ;  and,  having  placed  them  in  the  midway 
position,  1  put  a  lamp  under  each,  and  placed  a  thermometer 
with- its  ball  close  to  the  outside  of  the  case,  near  that  part 
which  one  of  the  weights  approached  to  in  its  positive  position, 
and  in  such  numner  that  1  conld  distinguish  the  divisions  by 
the  telescope.  Having  done  this,  I  shut  tlie  door,  and  some 
time  after  moved  the  weights  to  the  positive  position.  At  first, 
the  arm  was  drawn  aside  oidy  in  its  usual  manner  ;  but,  in 
half  an  hour,  the  effect  was  so  much  increased,  that  the  arm 
was  drawn  14  divisions  aside,  instead  of  about  three,  as  it 
would  otherwise  have  been,  and  the  thermometer  Avas  raised 
near  l^.f)  ;  namely,  from  (51"  to  (>''^°.5.  On  opening  the  door, 
the  weights  were  found  to  be  no  more  heated,  than  just  to  pre- 
vent their  feeling  cool  to  my  fingers. 

76 


TIIK    LAWS    OK    (JUAVlTA'iON 

As  tho  oiToot  of  ji  (litTcrcMicc  of  tcriiixM'iitiire  jippi-aiTtl  to  hv 
so  grt'iit,  I  bored  a  siiiull  liolo  in  oiiu  of  the  wri^Mits,  about 
tlirec-quiirters  of  jui  iiicb  (loop,  iind  iiisortod  tlio  bull  of  a  sriiull 
tliormomotor,  jiiid  thou  oovorocl  up  tlio  opouiuj,'  witii  ooiuont. 
Another  suuill  therniorueter  was  plaoed  with  its  ball  close  to 
tho  case,  aud  as  near  to  that  part  to  which  the  woi<,dit  was 
approached  as  could  be  done  with  safety  ;  the  theruioniotors 
being  so  i)laced,  that  when  the  weights  were  in  the  negative 
position,  both  could  be  seen  through  one  of  the  telesco])es,  by 
means  of  light  rellected  from  a  concave  mirror. 


or 
,rt 

>y 

18 

^t, 

in 
im 

lit 

d 


Exrr<:KFMKNT  \'I.     Skit.  0 

Wt'if//ilx  ill   miihriOf  jittsift'ori 


Tliornioinetor 

iTtI  romp 

Divi.sions 

Time 

I'oinl  of 

TjAI'I  l-IIK' 

points 

rust 

III   iilr 
55.5 

ill  \v{<i)jlit 

18.9 

9''   43 

18.85 

10       3 

18.85 

Weighta  moved  to  inyative  pmtion 

13.1 

— 

10     12 

— 

55.5 

55.8 

18.4 

— 

18 

15.82 

13.4 

— 

25 

missed 

13.6 

— 

39 

— 

55.5 

55.8 

17.6 

— 

46 

15.65 

13.8 

— 

53 

15.65 

17.4 

— 

11      0 

15.65 

14.0 

— 

7 

15.65 

17. 'i 

— 

14 

— 

55.5 

Weif/hta  moved  to  jiositive  jwsition, 

25.8 

— 

23 

17.5 

— 

30 

21.55 

25.4 

— 

37 

','1  .6 

18.1 

— 

44 

21   (i.-) 

25.0 

— 

51 

niissiid 

24.7 

— 

0      5 

19. 

— 

12 

21.77 

24.4 

— 

19 

M 

otiou  of  arm  on  moving  weights  from  midway  to  —  = 

3.03 

-  to  +  = 

5.9 

77 


■     rw '■■  ""T- 


M  KM  OIKS    UN 


0;      ii 


1:51 1 


ill 


MM: 


^;!ii 


i  '!">'' 


Extreme 
pnliilH 


13.6 

18.8 
13.8 


16.9 
14.5 
16.6 


26.4 
17.2 
26.1 


19.3 
25.1 
19.7 


EXFEUIMKNT    VII.      SkI'T.    18 
Weifjfitu  in  viuhnay  posit  ion 


DIVIHiOMK 


19.4 
19.4 


Time 


I'diiii  or 

rcHl 


Thertnomoler 
111   iiir  in  woiK>it 


Sh  30  —  56.7 

9     32  —  56.6 

Weif/ht«  mored  to  net/dtivc  poHilion 

40  - 

47  16.25 

54 

Eiijht  cjctveme  jvUiUh  niinHed 

10  58 

11  5  15.62 
12 

Weighta  mored  to  positive  position 


57.2 


20 

28 
35 


21.72 


56.5 


Four  extreme  points  missed 

0    10 

17  22.3 

24 


Motion  of  arm  on  moving  weights  from  midway  to  — =  3.15 

-  to  +  =  6.1 


ExPKRiMEivr  VIII.     Sept.  23 
Weights  in  midway  position 


Extreme 
points 


13.5 
18.6 
13.6 


17.4 
14.1 

17.2 


Divisions 

Time 

9"  46' 
10     45 

Tolnl  of 
rest 

Thermometer 

in   air 

in  weight 

19.3 
19.2 

19.2 

53.1 
53.1 

T 

Veights  mored  to  negative  position 

— 

56 

11      3 

10 

16.07 

53.6 

Four  extreme  points  missed 

— 

44 
51 

58 

15.7 

53.6 

78 


TlIK    LAWS    UK    <J1:AV1TATIUN 


Weiyhtn  vifyred  to  jMmitive  jumition 


15.7 
20.7 
16.0 


25.0 
18.1 
25.5 


0''     1 


i; 


21.42 


53.15 


Ttto  extreine  jiouitti  ininHetl 


:iO 
48 

50 


21.9 


Motion  of  arm  on  moving  wiii^lit.s  from  niiilwny  to  —  =  8.1!i 

-to+=:5.72 

In  these  tliree  experiinonts,  the  ctTect  of  tlie  weiglit  appeared 
to  inerease  from  two  to  live  tenths  of  a  (livisioii,  on  stanilin*; 
an  liour  ;  and  tlie  tljerniometers  shewed,  that  the  weights  were 
three  or  five  tenths  of  a  degree  wanner  tlian  tlu;  air  close  to  the 
case.  In  the  two  hist  experiments,  1  pnt  a  himp  into  the  room, 
over  night,  in  liopesof  making  the  air  warmer  than  tlie  weights, 
but  without  effeet,  as  the  lieat  of  the  weights  exceeded  tliat  of 
tlie  air  more  in  tliese  two  experiments  than  in  the  former. 

On  the  evening  of  Oc;tol)er  17,  the  weights  being  phieed  in 
tlie  midway  position,  lamps  were  pnt  under  them,  in  order  to 
warm  them  ;  the  door  was  then  shut,  and  the  lamps  suffered  to 
burn  out.  The  next  morning  it  was  found,  on  moving  the 
weights  to  the  negative  position,  that  they  were  7°. 5  warmer 
than  the  air  near  the  case.  After  they  had  continued  an  hour 
in  that  position,  they  were  found  to  have  cooled  l^.T),  so  as  to 
be  only  0°  warmer  than  the  air.  Tiiey  were  then  moved  to  the 
positive  position  ;  and  in  botii  positions  the  arm  was  drawn 
aside  about  four  divisions  more,  after  the  weights  had  remained 
an  hour  in  that  position,  than  it  was  at  first. 

May  22,  17!J8.  The  experiment  was  repeated  in  the  same 
manner,  except  that  the  lamps  were  made  so  as  to  l)urn  only  a 
short  time,  and  only  two  hours  were  suffered  to  elapse  before 
the  weights  were  moved.  The  weights  were  now  found  to  be 
scarcely  2°  warmer  than  the  case  ;  and  tlie  arm  was  drawn 
aside  about  two  divisions  more,  after  the  weights  had  remained 
an  hour  in  the  position  they  were  moved  to,  than  it  was  at  first. 

On  May  23,  the  experiment  was  tried  in  the  same  manner, 
except  that  the  weights  were  (tooled  by  laying  ice  on  them;  the 
ice  being  confined  in  its  place  by  tin  plates,  which,  on  moving 
the  weights,  fell  to  the  ground,  so  as  not  to  be  in  the  way.  «)n 
moving  the  weights  to  the  negative  position,  they  were  found 

79 


-u 


^^ 


MK  Mollis    ON 


S<1 


111 


I'^'^M 


to  bo  lUmiit  S''  ooldcr  tluui  the  air,  and  tlioir  rfTont  on  the  arm 
BocniLMl  now  to  (liininiHh  on  Hlandin;;,  instcud  of  iniM'oasin^,  as 
it  did  boforo  ;  as  tbc  arm  was  drawn  asido  about  *^J  divisions 
less,  at  tbo  ond  <>f  an  iionr  al'tor  tiio  motion  of  the  wnights, 
than  it  was  at  first. 

It  Kooms  snfliciontly  provcMl,  tbtM'cforo,  that  the  efTect  in  ques- 
tion is  produced,  as  above  (ixphiinod,  l)y  the  dilTeren(!e  of  tem- 
perature Ix'tween  tiie  wei«?lits  and  ease;  for  in  tiie  (Ith,  Htl).  and 
IHh*  experiments,  in  whieh  the  wei<;hts  were  not  much  warmer 
than  tile  case,  tiieir  (dfec^t  iiu^reased  but  little  on  standing; 
whereas,  it  increased  much,  when  they  were  much  warmer  than 
the  case,  and  decreased  fiuHsh,  when  they  were  much  cooler. 

It  must  be  observed,  that  in  this  a])))aratus,  the  box  in  which 
the  balls  play  is  pretty  deej),  and  the  balls  hang  near  the  bot- 
tom of  it,  whi(!h  makes  the  efTect  of  the  current  of  air  more 
sensible  than  it  would  otherwise  be,  and  is  a  defect  which  1  iu- 
tend  to  rectify  in  some  future  experiments. 


'11 


'', 


iiii: 


;iitl 


i:|!il 

;ii'(i 


•I  ill 


I 


ExpKiiiMENT  IX.     April  29 

Weightx  in  pimtlre  ixmtion 


Kxtromo 

Divi.sioiiH 

Time 

Toiiit  of 

Time  or  middle  of 

poiiils 

rt'sl 

vliinilioii 

34.7 

85. 

— 

— 

34.84 

34.05 

Weights  moved  to  negntive  p<mtio7i 

23.8 

28 
29 

11"  18'    39"  \ 

58    S 

— 

11"  18'   43" 

33.2 

— 

— 

28.52 

29 

28 

35    27    ) 
57    \ 

— 

25    40 

33.9 

— 

28.25 

33. 

— 

— 

28.01 

24.15 

— 

— 

37.82 

31. 

— 

— 

27.(53 

34.4 

— 

— 

37.55 

30.4 

28 

0      7      4    1 

53    S 

27.47 

27 

0      7    26 

24.7 

Motion  of  ivnn        —  6.32 

Time  of  vibnition  =:  6'  58" 

*  [^Tku  is  evidently  a  misprint  for  Qth,  1th,  and  8///.] 

80 


f 


T 11  !•;    1.  A  \V  S    ( )  K    ( J  K  A  \'  I T  A  T 1 U  N 


I 


Kxi'KUiMKNT  \.     May  5 


Rxtmine 

|Hllll(M 

a4.5 

;m.5 

84.4 


22.3 


88.2 


22.0 

Wi.Ti 

2:i.2 

31.45 

23.5 


30.7 

23.95 

80.25 


IHvIhIuUH 


2S 
29 

2S 
27 


27 

28 

28 
27 

27 

28 


Weif/htx  in  jxmiliir  jutnition 


Time 


I'liinl  or 
rvMl 


33.9^ 


TIlID'  1)1'  llilililli'  III' 

viliriilKiii 


Weiyhts  moral  to  myiitirf  jioxitiou 


10"  43  42  I 

44  0  ( 

r,{)  33  / 

51  0  S 


11     25    20  } 

58  )■ 

32      0  / 

32    40  S 

39  19  ) 

40  2  ] 


07    tJO 


27.72 
27.7 

27.58 
27.4 

27.28 


27.21 
27.21 


m  43    30 


50    30 


11  25  24 
32  27 
39    23 


Motion  of  arm       =0.15 
Time  of  vibration  =  0  59  " 


l»in«'r<'iic( 


7'     0 


7      3 
0    50 


EXFKHIMENT    XI.      MaY   6 

Wrights  in  jtom'tive  jxmtion 


Extreme 
puiutH      • 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

34.9 

34.1 

— 

— 

34.47 

34.8 

— 

— 

34.49 

34.25 

Weights  maved  to  negative  position 

23.3 

28 
29 

9''   59'    59    } 
10      0    27    S 

— 

lO"     0'     8' 

33.3 

— 

— 

28.42 

29 
27 

6  52    ) 

7  51    S 

— 

7      5 

23.8 

— 

— 

28.35 

F 

81 

^. 


-f^^-,- 


^MAGE  EVALUATION 
TEST  TARGET  (MT-3) 


^  5.^t% 


^>  M^ 


m 


y. 


:/. 


4C 


% 


1.0 


1.1 


|io   "'^"     mH 

■^  ^    ■2.2 
2.0 


us 

■It 


IL25  111  1.4 


I 

M 
1.6 


^ 


V] 


/ 


^7 


"^S 


y 


Hiotographic 

Sciences 

Corporation 


23  WEST  MAIN  STREET 

WEBSTER,  N.Y.  MS80 

(716)l7'i-4S03 


V 


iV 


H^ 


■\ 


'<^ 


1 


MEMOIUS    ON 


82.5 

__ 

— 

28.3 

24.4 

missed 

24.8 

81.3 

— 

— 

28.17 

29 

28 

10"  48'    37"  ) 
49    21    \ 

— 

10"  49'     8" 

25.3 

— 

— 

28.2 

28 
29 

56      8    I 
56    \ 

— 

56    13 

30.9 

Motion  of  arm       =6.07 

Time  of  vibralion  =  7'  1 " 

In  the  three  foregoing  experiments,  the  index  was  purposely 
moved  so  that,  before  the  beginning  of  the  experiment,  the 
balls  rested  as  near  the  sides  of  the  case  as  they  could,  without 
danger  of  touching  it ;  for  it  must  be  observed,  that  when  the 
arm  is  at  35,  they  begin  to  touch.  In  the  two  following  ex- 
periments, the  index  was  in  its  usual  position. 

Experiment  XII.     May  9 

Weif/htK  ill,  negative  position 


Extreme 
points 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  of 
vibration 

17.4 

9"  45      0 " 

17.4 

58      0 

17.4 

10      8      0 

17.4 

10      0 

17.4 

Weights  moved  to  positii 

'e  position 

28.85 

24 
22 

20  50    ) 

21  46    ( 

— 

10"  20'   59" 

18.4 

— 

— 

23.49 

28.3 

— 

23.57 

• 

19.3 

— 

— 

23.67 

• 

27.8 

— 

— 

23.72 

20. 

— 

— 

23.8 

27.4 

— 

.  — 

23.83 

24 
28 

11      3    13    ) 
54    \ 

— 

11      3    14 

20.55 

— 

— 

23.87 

28 
24 

9    45    I 

10    28    f 

— 

10    18 

87. 

Motion  of  arm        = 

:6.09 

Time  of  vihrulinn  = 

:7'3' 

88 


8" 


18 


29.6 
17.4 
28.9 

18.4 


28.4 
19.3 

27.8 


19.9 

27.3 

13.5 
21.8 

13.9 


21.1 
14.4 
20.5 
14.7 
20. 


15. 


19.5 


THE    LAWS    OF    GRAVITATION 
Experiment  XIII.     May  :e5 

WeifjhtH  in  tm/ntive  poHition 


25 
?4 

23 
24 

24 
23 

23 
24 


24 
23 


17.2 

Weights  moved  to  positive  position 
lOh  22'   22" 


22"  ) 
45    f 

29  59    ; 

30  23    f 

36  58    1 

37  24   / 


44 


i\ 


11 


23 
24 


5    26 
6 


u 


12     12 


23.32 

23.4 

23.52 


23.63 

23.7 

23.7 


23.72 


18 
17 

17 
18 


50    } 

Weights  moved  to  negative  position 

17.75 


37  34    ) 

38  10    f 


44    26    ) 

4    f 


45 


18 
17 

17 
18 


0    19    57    ) 
20    52    \ 


27  15 

28  15 


\ 

Motion  of  the  arm  on  moving  weights 
Time  of  vibration  at  + 


17.67 


17.62 

17.6 

17.52 

17.47 

17.42 


17.37 


Time  of  middle  of 
vibration 


10"  22'  56" 

30  3 

37  7 

44  14 

11   5  31 
12  35 


37  39 
44  45 


0  20  24 
27  30 


from- to +  = 
+  to  -  = 


83 


6.12 
5.97 

:7'6" 

:7'7" 


! 

MEMO  ins    UN 

• 

i                                                      ExpKRiMKNT  XIV.     May  26 
I                                                                         Wcif/htM  in  nef/afire  position 

•                                             Kxlromo 
|>                                                           imintH 

DIvisiuiiM 
16.1 

Time 

9''    IH       0" 

Point  ol 
rest 

TImo  of  middle  n( 
vibriition 

16.1 

24       0 

16.1 

46      0 

16.1 

49      0 

16.1 

Wei(/htH  moved  to  positive  jnysition 

27.7 

i:^^ 

23 
22 

10      0    46    ) 
1     16    ^ 

— 

10"   r    1" 

^-:                                           17.3 

— 

— 

22.37 

22 

7    58    / 

8      5 

23 

8    27    f 

,..;:                                                27.2 

— 

— 

22.5 

23 
22 

15      2    ( 
32    i 

— 

15      9 

18.3 

— 

— 

22.65 

26.8 

— 

— 

22.75 

19.1 

— 

— 

22.85 

26.4 

— 

— 

22.97 

-. 

23 
22 

43  40    } 

44  22    f 

— 

43    32 

20. 

— 

— 

23.15 

22 
23 

49  53    ) 

50  37    \ 

— 

50    41 

26.2 

Weiijhts  moved  to  negative  position 

12.4 

16 
17 

11      7    53    ) 
8    27    S 

— 

11      8    25 

21.5 

— 

— 

17.02 

17 
16 

15  30    I 

16  3    \ 

— 

15    27 

12.7 

— 

— 

16.9 

uiT                                   20.7 

— 

— 

16.85 

Ih!                                    13.3 

— 

16.82 

'^'•'"^                                    20. 

— 

— 

16.72 

13.6 

— 

— 

16.67 

16 
17 

50  33    ) 

51  19    f 

— 

50    58 

19.5 

— 

— 

16.65 

■ 

17 
16 

57  53    I 

58  44    \ 

— 

58      6 

14. 

Motion  of  arm  by  moving  weights  from  —  to  + : 

=  6.27 

+  tO-: 

=  6.13 

j :                                                Time  of  vibratioi 

1  at  -f 

=  7'  6" 

■1.1'                                               »                                                                                                                       """ 

=  7'  6" 

84 

■  J     ■'•'-■■  'I        \ 

1  1    ■  .  1   ." 
'  ■    ',1 1 

THE    LAWS    OF    (JUAVITATlUN 


J5 
J7 


r 

6 


III  the  next  oxperinieiit,  the  balls,  before  the  motion  of  the 
weights,  were  inude  to  rest  as  near  as  possible  to  the  sides  of 
the  case,  but  on  the  contrary  side  from  what  they  did  in  the 
i)th,  loth,  and  11th  experiments. 

ExfKiiiMKNT  XV.     May  27 

WiUjhtx  in  negative  jxtnilion 


Extreme 
puiiits 

Divisions 

Time 

I'oint  of 
res  I 

Time  of  middle  of 
viliriitioii 

8.9 

8.85 

— 

— 

8.61 

3.85 

— 

^— 

8.61 

3.4 

Weights  moved  to  poxiti 

ve  jHitiitioii 

15.4 

10 
9 

10"    5'  59"  ) 

6  27    \ 

— 

10"    5'  56" 

4.8 

— 

— 

9.95 

9 
10 

12   43    I 
18   n    f 

— 

13     5 

14.8 

— 

— 

10.07 

10 
9 

20   24    } 
56    f 

20   13 

5.9 

— 

— — 

10.23 

14.35 

— 

— 

10.85 

6.8 

— 

— 

10.46 

18.9 

— 

— 

10.52 

11 
10 

48  80    1 

49  11    \ 

— 

48  42 

7.5 

— 

— 

10.6 

10 

11 

55  26    ) 

56  10    \ 

— 

55  48 

13.5 

Motion  of  tlie  arm  — 

6.34 

Time  of  vibration  = 

7'  7" 

The  two  following  experiments  were  made  by  Mr.  Gilpin, 
who  was  so  good  as  to  assist  me  on  the  occasion. 

Experiment  XVI.     May  28 

Weights  in  negative  j)osition 


Extreme 
points 

UivisioDS 

Time 

Point  or 

rest 

Time  of  middle  of 
vibration 

22.55 
8.4 

21. 
9.2 

— 

— 

15.09 
14.9 

85 


MEMOIRS    ON 


Wei(/fits  moved  to  jwintive  position 


26.6 

22 

ai 

10"  22   53    ) 
23   20 

— 

lOh  23'  15 ' 

15.8 

— 

— 

21. 

20 
21 

36 

— 

30  30 

25.8 

— 

— 

21.05 

32 
21 

37   23    I 
55    \ 

— 

37  45 

16.8 

— 

— 

21.11 

20 
21 

44  29    I 

45  4 

— 

45     1 

25.05 

— 

— 

21.11 

22 
21 

51  54    / 

52  32    ] 

— 

52  20 

17.57 

— 

— 

21.2 

21 
22 

59   31    / 
11     0    13    f 

— 

59  34 

24.6 

— 

— 

21.28 

22 
21 

6  24    1 

7  9    f 

— 

11     6  49 

18.3 

Motion  of  the  iirrn  : 

=  6.1 

T 

irne  of  vibration  : 

=  7'  16" 

13  !M!: 


,,    -if!'  ?. 


Experiment  XVII.     May  30 

Weights  in  negative  position 


Extreme 
poiutH 

Divisions 

Time 

Point  of 
rest 

Time  of  middle  or 
vibration 

17.2 

10"  19'    0" 

17.1 

25     0 

17.07 

29     0 

17.15 

40     0 

17.45 

49     0 

17.42 

51     0 

17.42 

11      1     0 

17.42 

Weights  moved  to  positive  position 

28.8 

24 
23 

11    11   23    ) 
49 

— 

ll""  11'  87" 

18.1 

— 

— 

23.2 

22 
23 

18   13    / 
43 

— 

18  42 

27.8 

— 

— 

23.12 

24 
23 

25   19    ) 
49 

— 

25  40 

18.8 

— 

— 

23.2 

23 
24 

32  41    ) 

33  13    y 

86 

— 

32  43 

.'.iv».J*!i.;,v.:--:=Vr 


..^■...i-:;.sf^.^>,.. 


^.Hs: 


27.38 

19.7 

37. 

20.4 

36.5 

20.8 

36.25 

13.3 

22.4 

13.7 

21.6 

14. 

20.8 

14.3 

20.1 

14.6 


TJIK    LAWS    OF    (JKAVITATION 

23.31 


24 
23 

23 
24 

24 

33 

28 

24 

24 
23 

23 
24 


11"  30   28    ) 
40     3    J 

46  33    ) 

47  11    f 

53  36    ) 

54  17    f 

0     0   34    / 

1    18    f 

7  34    } 

8  21    f 

14  30    ) 

15  24    f 


23.44 

23.53 

33.57 

23.55 

23.59 


Weights  fmved  to  neffative  position 


17 

18 

18 
17 

17 

18 

18 
17 

17 

18 

18 
17 

17 

18 

18 
17 


32  19  / 

48  f 

— 

39  46  ) 

17.95 

40  19  f 

— 

46  26  ) 

17.85 

.47  0  f 

-^                   17.72 
53  43  } 

54  20  f      — 

.   0-89  ,     ''■' 

1  30  f      - 

7  39  ) 

17.47 

H  31  f 

: — 

14  54  / 

17.37 

15  43  f 

— 

31  33  ) 

17.27 

33  33  )" 

— 

11"  39  44' 
46  46 
53  48 
0    0  55 
7  50 
14  58 


32 

44 

39 

44 

46 

48 

53 

50 

1  0 

55 

7 

59 

15 

4 

22     5 


Motion  of  the  ann  on  moving  weigh.s  fn.m  -  to  +  :.  5.73 

Time  of  vibmtion  at  +  +  to  -  =  5  64 

=  7'  2" 

=  r  3" 


MKMOlltS    UN 
On  thk  Mktiioh  ok  C'ompitino  tiik  Dknsity  av  Tirn  Eauth 

I"  IK )  M    '!•  1 1  i;s  !•;    K  X  !•  K  K I  M  K  N  IS 

I  sliiill  first  ooriipiito  tluH,  on  tlio  supposition  that  tlio  arm 
iuul  copper  rods  luive  no  wciglit,  and  that  the  \voi«i:lits  exert  no 
sensible  attra(!tion,  ex(X'pt  on  tlie  nearest  ball  ;  and  shall  then 
examine  what  corrections  are  necessary,  on  account  of  the  arm 
and  rods,  and  some  other  small  ctiuses. 

The  first  thing  is,  to  find  the  force  required  to  draw  the  arm 
aside,  which,  as  was  before  said,  is  to  be  determined  by  the  time 
of  a  vibration. 

The  distaiKse  of  the  centres  of  the  two  balls  from  eacdi  other 
is  73. IJ  inches,  and  tlierefore  the  distance  of  each  from  the 
centre  of  motion  is  IJfi.Cr),  and  the  length  of  a  pemluluni  vi- 
brating seconds,  in  this  climate,  is  3!).l-t;  therefore,  if  the 
stiffness  of  the  wire  by  which  the  arm  is  suspended  is  such, 
that  the  force  which  must  be  applied  to  each  ball,  in  order  to 
draw  the  arm  aside  by  the  aiigle  A,  is  to  the  weight  of  that 
ball  as  the  arch  of  A  to  the  radius,  the  arm  will  vibrate  in  the 
same  time  as  a  pendulum  whose  length  is  3G.G5  inches,  that  is,  in 

V  ')n  I  1  f^cconds  ;  and  therefore,  if  the  stiffness  of  the  wire  is 

such  as  to  make  it  vibrate  in  N  seconds,  the  force  which  must 
be  applied  to  each  ball,  in  order  to  draw  it  iiside  by  the  angle 

A,  is  to  the  weight  of  the  ball  as  the  arch  of  Ax^.,x'.  ',' 
*^  ^      ;V.J.14 

to  the  radius.      Hut  the  ivory  scale  at  the  end  of  the  arm   is 

38.3  inche.".  from  the  centre  of  motion,  and  each  division  is  gij, 

of  an  inch,  and  therefore  subtends  an  angle  at  the  centre,  whose 

arch  is  ^^^  ;  and  therefore  the  force  which  must  be  applied  to 

each  ball,  to  draw  the  arm  aside  by  one  division,  is  to  the  weight 

of  the  ball  as 


7G0N* 


30.05  ,     ,  1       i.    1  * 


*  [Or  thitu:  using  the  ordinary  notation  for  the  simple  lendulum  vibrating 
through  small  arcs,  if  the  force  on  each  ball  drawing  the  arm  aside  through  an 


arc  subtending  an  angle  of  A°  were  mg  x 


arc 
radius 


,  tlie  arm  would  vibrate  like  a 


jtendulum  of  the  same  length,  and  have  a  period 


seconds,  because  the 


period  of  a  pendulum  varies  as  tfie  squarz  root  of  its  length.    But  the  force  varies 
as r-T-j ;  therefore  tJie  force  required  to  draw  the  arm  through  A°  with 


{periodf 


88 


Til  K    LAWS    OK    (illAV  ITATloN 

Tlio  next  tiling  is,  ^o  tiixi  tlio  proportion  wliicli  tlie  iittraction 
of  the  wcij^ht  on  tlio  hail  hciirs  to  that  of  the  carlli  thi'roon,  sup- 
posini;  the  i)all  to  hv  place*!  in  the  middle  of  tlic  case,  that  is, 
to  he  not  ncarcM*  to  one  side  than  the  other.  When  the  \veij,'ht8 
are  approached  to  tlie  hiills,  their  centres  are  S.S.")  inches  from 
the  middle  line  id"  tlit?  ejise  ;  hnt,  thronjifh  inadvertence,  the 
distance,  from  eacdi  other,  of  the  rods  whi(di  snpport  these 
weif^hts,  was  made  ecpial  to  the  distaiu^e  of  the  centres  of  the 
halls  from  each  other,  whereas  it  ou^ht  to  have  heen  somewhat 
greater.  In  conse(juence  of  this,  tin;  centres  of  the  wei<;hts  are 
not  exactly  opposite  to  those  of  the  halls,  when  they  are  ap- 
proached together;  and  the  effect  of  the  weights,  in  drawing 
the  arm  aside,  is  less  than  it  wonid  otherwise  have  heen,  in  the 

triplicate  ratio  of  '  ''  '    to  the  chord  of  the  an<:le  whose  sine  is 

u  t,r  .)o.M,) 

.T;r-'!r'  or  ill  the  trii)li(  ate  ratio  of  the  cosine  (d'  i  this  angle  to 
ob.bo 

the  radius,  or  in  the  ratio  of  .977!)  to  1.* 


diosc 
kI  to 
sight 


\'ating 
jhan 

like  a 


se  the 

Uivies 

with 


.   ,  „„  are  of  A"      ^0.6.") 


N*.     And  the  force  required  to  draw 


the  arm  through  )  .scale  divmoti  with  period  N" 

36.65     1 

=  ^^^-36  65     ^39.14^^ 


*  [lA!t  W  be  tile  position  of 
the  ''weight  "  of  mans  W,  B 
the  position  it  was  intend- 
ed that  it  should  hate,  and  m 
that  of  the  "ball"  of  mass  m. 
The  distance  mB,  or  WA,  is 
8.S5  inches,  and  OVV  and  Om 
36.65  inches.  Call  WA  and 
Wm  a  and  b  resjiectively,  and 
G  the  gravitation  constant. 
Then  it  was  intended  that 
tJie  attraction  to  move  the  arm 

should  be    ; — ,  but  it  is 

//* 

GWm    a 

was  intended  in  the  ratio  of 


36.65 


39.14-  '"^  ""  H18N»  1 


B    W. 


a' 


;  and  so  is  less  than 


a 


6 


,3  to  1,  or  of  Oos^  ^  to  1.] 


Fig.  d 


!''  'f\ 


il!!i 


V 


|i-; 


i|  , 


;   H 


M  I:M(H  lis    ON 

Viiich  of  tlio  uoijjlits  woiijlis  'l,A:V.),(H)i)  fjniins,  and  tlioroforo  is 
0(jiihI  ill  \v('i,i,Hit  to  10.(14  splicricul  l'i»ot  of  wiitor  ;*  jiiid  tlu-nifore 
its  iitti'iKiinii  on  a  piiiticlc  pluccd  at,  tlit'  ('ciitroof  tlio  ball,  is  to 
tlie  attraction  of  a  spliorical  l'(»ot  of  watcM-  on  an  (Mjiial  jiarticlo 

piacod    on    its   siirfa<'i%  as   10.04  x  .I'iT'.t  X  (  ,   ,.  )    to  1.     Tho 

mean  dianiotcr  of  tiic  oartli  is  41,S(K»,(MMI  IVt't  ;f  and  tliorefore, 
it  the  mean  density  of  the  earth  is  to  that  of  waiter  as  1)  to  one, 
the  attraction  of  the  leaden  weii^hton  the  hall  will  he  to  that  of 


tlu!  earth  thereon,  as  1().«>4  x.Oi^Dx 
1  •  8,;;}!),(K)(U).t 


\S.S,»/ 


lo  41,80(),0(K)  I)  : 


*  [  T/iii/  is,  t's  (f/Kiil  to  tlic  weifiht  ofn  xjifieir  of  water  which  ran  he  iiiHcrihed  in 
a  I'lihc  whose  nil  nine  in  10.64  en.  ft.,  or  n^e  ean  erpretm  the  rolinne  of  the  sphere 
/>!/  the  unni/ter  10.04,  when  the  nnit  of  rolnnie  is  that  of  a  sphere  of  \  foot  in. 

diameter,  that  is,  of —  en.  ft     The  radins  of  a  spherical  f<H>t  of  water  is,  aecf>rd- 

inf/lt/,  6  inches.     (\irendi.sh  ecidtntli/  nses  Kirwan's  estimate  (fiT^W.'^Tt  f/rainH 
to  the  CH.  in.  of  wafer. 

The  ensnin;/  ca/cnfation  can  he  stated  thns:  ('<dl  o'  and  d'  the  densities  of 
water  and  (f  the  earth  respect iceh/,  m  the  mass  ff  the  hall,  and  (»  the  r/ra cita- 
tion constant.  The  coin  me  of  the  earth  in  spherical  nnifs  is  (41  800  000)'.  and 
its  radius  Qx 41  800  000  inches. 

Gx  lOMxdxm 

Aft  met  ion  of  iceifiht  on  hall  at  8.85  inches (8.8r))- 

Attraction  of  earth  on  ball 


X  9779 


Gx(41  80(MM)0)'xr/'x  m. 
(6x41800  000)' 

.9779  x  10.64  xf-^^_Y 
\H.8.v 


But  we  have  already  fov nd  {par/e  89) 
Forc^  required  to  draw  the  arm  fhrout/h  1  div. 


Weight  of  hall 

Diridinfi  equation  (1)  by  (2)  we  have 
Attraction  tf  Wtif/ht  on  hall 


41  800  000  ~ 
a 

1 

'8  7;jy  000  D 

.  .  .  . 

(1) 

1 

818N«  •  • 

.  .  . 

(3) 

818  N2 

N2 

8  739  GOOD  ~ 

"  10  6831) 

Force  required  to  draw  the  arm  throuf/h  1  die. 

=  no.  of  die.  throvgh  which  th^  arm  i,i  dnurn~li  div.] 

f  III  strictii(!Ks,  we  oiigiit,  inslcad  of  tlie  nieiiti  diameter  of  the  earth,  to 
take  the  (liiiineter  of  that  sphen;  whose  aitiiiction  is  equal  to  the  force  of 
gravity  in  this  climnte;  but  the  difference  is  not  worth  regarding. 

X  [JIutton  has  pointed  out  (54)  that  this  numl)er  should  he  8,740,000;  but  it 
will  not  make  any  appreciable  change  in  the  value  of  D.] 

90 


.')li 


Til  K    LAWS    (>!•    (i  ItA  V  IT  ATION 


«)' 


0) 


(2) 


1,  to 

be  of 

\ut  it 


It  is  slunvii,  tlicn'foro.  that  tlic  force  uliich  miiHt  Ix;  applied 
to  oju'li   ball,  in  order  to  dniw  tjie  arm  oiu    division  out  of  its 

natural  position,  is  ..^  of  the  weiijlit  of  the  hall  ;  and.  if  the 

mean  density  of  tlie  earth  is  to  that  of  water  as  I)  to  I,  I  ho  at- 

tra(!tion  of  the  wjMi;ht  on  the  hall  is  — ^;      MMi>  u  **'  ''"'  ^V('i«^'lit  (»f 

that  hall  ;  and  therefore  the  attraction  will  he  ahle  to  draw  the 

arm  out  of  its  jiatnral  position  by     ^,    '    '      ..or,-  ,  ^  divis- 

ions  ;  and  therefore,  if  on  movin<(  the  weii^hts  from  tlm  mid- 
way to  a  near  position  the  arm  is  found  to  move  h  divisions,  or 
if  it  moves  'Z  B  divisions  on  nn)ving  the  wei;,dits  Irom  one  near 
position  to  tho  other,  it  follows  that  the  density  of  the  earth. 

We  must  now  consider  the  corrections!  which  Tuust  be  ap- 
plied to  this  result  ;  Hrst,  for  the  ert'ect  which  the  resistance  of 
the  arm  to  motion  has  on  the  time  of  the  vibration  :  t>d,  for  the 
attraction  of  the  weights  on  the  arm  :  )M,  for  their  atirac^tion 
on  the  farther  ball  :  4th,  for  the  attraction  of  the  copper  rods 
on  the  balls  and  arm  :  r)th,  for  the  attracttion  of  the  case  on  the 
bulls  and  arm  :  and  0th,  for  the  alteration  of  the  attraction  of 
the  weights  on  the  balls,  according  to  the  position  of  the  arm, 
and  the  effect  which  that  has  on  the  time  of  vibration.  None 
of  these  corrections,  indeed,  except  the  last,  are  of  much  sig- 
nification, but  they  ought  not  entirely  to  be  neglected. 

As  to  the  first,  it  must  be  co!isidered,  that  during  the  vibra- 
tions of  the  arm  and  balls,  part  of  the  force  is  s])ent  in  acceler- 
ating the  arm  ;  and  therefore,  in  order  to  find  the  force  re- 
quired to  draw  them  out  of  their  natural  position,  we  must 
find  the  proportion  which  the  forces  spent  in  accelerating  the 
arm  and  balls  bear  to  each  other. 

Let  EDC«/6'  (Fig.  4)  be  the  arm.  B  and  <'j  the  balls.  C.s- 
the  suspending  wire.  The  arm  consists  of  4  parts  ;  first,  a  deal 
rod  l)cd,  73.3  inches  long ;  :^d,  the  silver  wire  \)Cd,  weighing  ITO 
grains  ;    3d,  the  end  pieces   DE  and  ed,   to  which  the  ivory 

*  [This  nvmber  should  be  10,085.     See  lust  note.] 

f  [For  a  discussion  of  these  corrections,  similar  to  that  of  Cavendish,  but 
with  modern  mathematical  treatment,  see  Keirh  (07).] 

91 


mi: Mollis  ON 


:  a^^l 

Wm 

I^B' ' 

E 

c 

A 

D 

a 

\ 

B 

I 

voniior  ih  fastoiiod,  oiicli  of  wliinh  wei^jlis  45  p;rjiiiiH  ;  luul  4tli, 
Homo  bnisH  work  iU',  at  tiio  et'utro.  The  doiil  rod,  whoii  dry, 
w('i«^lis  )l'.\'i()  grains,  but  wlion  very  diimp,  sis  it  ooiimioidy  wais 
during  tiie  exporirneiits,  woiglis  '^400  ;  tlio  trjuis\orse  seotioii  is 

of  tlio  slisipi!  roproscMited  in  Fig.  5  ;  the  tlii(^k- 
noss  liA,  and  the  dimensions  of  tlie  ])art 
1)1W,  Itt'ing  till'  same  in  all  j)arts  ;  but  the 
breadth  \M/  diminishes  gradually,  from  the 
middle  to  the  ends.  The  area  of  this  see- 
tion  is  .133  of  a,  square  inch  at  the  middle,  and  .140  at  the 
end;  and   therefore,  if  any   point  x  (Fig.  4)   is   taken   in   cd, 

and  — ,  IS   called   .r,  this   rod  weighs  --—  per   inch   at 

the  muUUe;  ^^.,^-^,  at  the  end,  mul   j^x    -;^^^       = 
at  :r;  and  therefore,  as  the  weight  of  the  wire  is 


Fig.  5 


7:j.;j 


170 
73.3 


per  inch,  the  deal  rod  and  wire  together  may  be  con- 


.-       ,               ,     ,             .   ,,    ,        341)0-1848  a;         .     . 
sidered  as  a  rod  whose  weight  at  x=      ~,y^n P^^'  i»ch. 

But  the  force  required  to  accelerate  any  quantity  of  matter 
placed  at  x,  is  proportional  to  x^ ;  that  is,  it  is  to  the  force  re- 
quired to  accelerate  the  same  quantity  of  matter  placed  at  d  as 
a;'  to  1  ;  and  therefore,  if  cd  is  called  /,  and  x  is  supposed  to 
flow,  the  fluxion  of  the  force  required  to  accelerate  the  deal 

92 


fili; 

■liiUii 


Til  K    l,A\VS    (M-    (IKA  V  I  TAT  ION 


-|C 


at 


X 


ire  18 


con- 


itter 

re- 

\d  us 

to 

Ideal 


rod  iUid  wire  is  proportioiiiil  to 


tlie  IliK'tit 


of  wliicli,  iit'iicratt'il  wliil«'  ./•  flows  from  r  to  d. 


I 


r.VVM)     [.^4S\ 


{:)0 


so  tliiit  the  force  required  to  luieelenite  eucli  lialf  of  the  deal 
rod  and  wire,  is  the  sjiiiie  ais  is  retjuired  to  u(!eelerute  ;$.'»(►  i^nuiihs 
phieed  at  d. 

'I'he  resistauee  to  tn«)tion  of  each  of  the  pieces  di\  is  e(|iial  to 
tliat  of  4S  j^M-ains  phiced  at  ^/ ;  as  the  distance  of  their  centres  of 


jrravitv  from  ('  is  ;{S  inche.- 


Th((  resistance!  of  tln^  l»rass  work 


at  the  centre  nniy  l)e  disref!;ardc(|  ;  ant!  tlierefore  the  whoh'  Uwvv 
refpiired  to  acceh;rate  the  arm,  is  tiie  same  as  that  re(|nired  to 
accelerate  :)tlS  grains  phiced  at  cAvh  of  the  points  I)  and  d. 

Kach  of  tlie  balls  weighs  11. "^O^  ;,n'ains,  and  they  art^  phuM'd 
at  the  same  distance!  from  the  (sentre  as  I)  ami  d;  ami  there- 
fore, the  force  required  to  accelerate  the  halls  ami  arm  to- 
f^cther,  is  the  same  as  if  each  hall  vveijifhed  II, <!»;(>,  and  the  arm 
had  no  weight;  and  therefore,  supposing::  the  time;  of  a  vibra- 
tion to  be  given,  the  force  required  to  draw  the  arm  aside,  is 
greater  than  if  the  arm  had  no  weight,  in  the  i)roportion  of 
ii,(;(;o  to  \\,WZ,  or  of  I.o;ja3  to  1. 

To  find  the  attraction  of  the  w  'ights  on  the  arm,  through  d 
draw  the  vertical  plane  dwb  perpendicular  to  D^/,  and  let  tn 
be  the  centre  of  the  weight,  which,  though  not  accurately  in 
this  plane,  nuiy,  without  sensible  error,  be  considered  as  placed 
therein,  and  let  b  be  the  centre  of  the  ball;  then  wh  is  hori- 
zontal and =8.85,  and  ^/6  is  vertical   and  =5. 5;  let  wd=a,  v)b 

dx 
z=.b,  and  let  ~,  or  \—x,  =  z;  then  the  attraction  of  the  weight 

on  a  particle  of  matter  at  x,  in  the  direction  bw,  is  to  its  at- 
traction  on  the  same  particle  placed  at  h:\b^ :  {a^-^-z^ry,  or  is 

proportional  to  ;,,  and  the  force  of  that  attraction  to 

h''x{l—z) 
move  the  arm,  is  proportional  to  ^^ -,  and  the  weight  of 

{(t^-^-z'TY 

the  deal  rod  and  wire  at  the  point  x,  was  before  said  to  be 

3490-1848  .r     1G42+I848z  .     ,  ,    ..       *  -^  ^ 

— — = -~- per    inch;   and    therefore,  if  dx 

7o.u  io.o 

flows,  the  fluxion  of  the  power  to  move  the  arm 

93 


ni^ 


on 


r  He 


I    'i': 


gurg^j 

1 

\ 

1 

* 

( 
1 

i 

liJ 

M KM  OIKS    ON 

=  hx  -^  ~ X ~=::ZX  (821+024  z)  X ^^ 1 

/rzxh'il-hU):\zV^y^j 


h^zx(H'n■^^^)'^z-9Uz') 

924//2x('j'  +  ^') 


(a'-^-rz'V^ 


a' 


;  wliicli,  as  -^^  =  .08, 


{a-'-\-Pzy 
J>'-'i  X  (80') +  1  (>;{;<;) 0:>4  h'z_ 

805///  io:w>'        low/    024 /»' 


The  fluent  of  this 


Fa 


P 


log 


a 


and  the  force  with  wliioh  the  attraction  of  the  weight,  on  the 
nearest  half  of  the  deal  rod  and  wire,  tends  to  move  the  arm, 
is  i)roportional  to  this  fluent  generated  while  z  flows  from  0 
to  1,  that  is,  to  128  grains. 

The  force  with  wliich  the  attraction  of  the  weight  on  the  end 

piece  de  tends  to  move  the  arm,  is  proportional  to  47  x  -3  [npproxi- 

mairlt/],  or  20  grains;  and  therefore  the  whole  power  of  the 
weight  to  move  the  arm,  by  means  of  its  attraction  on  the  near- 
est part  thereof,  is  equal  to  its  attraction  on  157  grains  placed 

157 
at^,  which  i^'      '  .  ,  or  .0130  of  its  attraction  on  the  ball.* 

It  must  be  observed,  that  tiie  effect  of  the  attraction  of  the 
weight  on  the  whole  arm  is  rather  less  than  this,  as  its  attrac- 
tion on  tlie  farther  half  draws  it  the  contrary  way;  but,  as  the 
attractiou  on  this  is  snuiU,  in  comparison  of  its  attraction  on 
the  nearer  half,  it  may  be  disregarded. 

The  attraction  of  the  weight  on  the  furthest  ball,  in  the 
direction  biv,  is  to  its  attraction  on  the  nearest  ball  ::  wb^ :  tvh'j; 
::.0017;1;  and  therefore  the  effect  of  the  attractioi.  of  the 
weigiit  on  both  balls,  is  to  that  of  its  attraction  on  the  nearest 
ball::. 9083  : 1. 


*  [A  few  minor  misprints  in  the  last  two  paragraphs  in  the  oi'iginal  paper 
have  been,  corrected.  A  recalculation  seems  to  give  142.5  instead  of  128,  and 
28  instead  of  29,  grains;  (his  would  change  the  valve  of  J}  hy  1  jmrt  in  1000.] 

f  [^This  is  erroneously  printed  in  the  oi'iginal  as  icd^ :  icW.] 

94 


TIIK    LAWS    OK   (JRAVITATION 


the 


the 

the 
Lrest 


To  find  tlie  attraction  of  the  copper  rod  on  the  nourost  hall, 
let  b  and  w  (Fig.  (J)  he  the  centres  of  the  hall  and  weight,  and 
ea  the  perpend icnlar  part  of  the  copper  rod,  which  consists  of 
two  parts,  ad  and  dc.  ad  weighs  :^2,0(/()  grains,  and  is  l(j  inches 
long,  and  is  nearly  bisected  by  w.  de  weighs  41, (MM),  and  is  4(1 
inches  long,  wh  is  8.85  inches,  and  is  perpendicular  to  ew. 
Now,  the  attraction  of  a  line  eWy 
of  uniform  thickness,  on  b,  in  the 
direction  bw,  is  to  that  of  the  same 
quantity  of  matter  placed  at  w  ::  bw 
'.eb;  and  therefore  the  attraction 
of    the    part    da    equals    that    of 

r, ,  or  1G,-J00,  placed  at  w ; 

db 

and    the   jittraction    of    de  equals 

that    of    41,000  x''"'xv^- 41,000 

I'd      be 

X — rX,  ,>  or  2500,  placed  nt  the 
ed     bd 

same  point ;  so  that  the  attraction 

of  the  perpendicular  part  of  the   w\ \h 

copper  rod  on  b,  is  to  that  of  the 

weight  thereon,  as  18,800  :  2,439,- 

000,  or  as  .00771  to  1.     As  for  the 

attraction  of  the  inclined  part  of 

the  rod  and  wooden  bar,  marked  ng.  c 

Pr  and  rr  in  Fig.  1,  it  may  sa^'ely 

be  neglected,  and  so  may  the  attraction  of  the  whole  rod  on  the 

arm  and  farthest  ball ;  and  therefore  the  attraction  of  the  weight 

and  copper  rod,  on  the  arm  and  both  halls  together,  exceeds  the 

attraction  of  the  weight  on  the  nearest  ball,  in  the  proportion 

of  . 9983 -}-.0139  +  . 0077  to  one,  or  of  1.0199  to  1. 

The  next  thing  to  be  considered,  is  the  attraction  of  the-  ma- 
hogany case.  Now  it  is  evident,  that  when  the  arm  stands  at 
the  middle  division,  the  attractions  of  the  opposite  sides  of  the 
case  balance  each  other,  and  liave  no  power  to  draw  the  arm 
either  way.  When  the  arm  is  removed  from  this  division,  it  is 
attracted  a  little  towards  the  nearest  side,  so  that  the  force  re- 
quired to  draw  the  arm  aside  is  rather  less  than  it  would  other- 
wise be  ;  uut  yet,  if  this  force  is  proportional  to  the  distance  of 
the  arm  from  the  middle  division,  it  makes  no  error  in  the  re- 

95 


\l 


il 
,  '1? 


n 


"I 

1 


'HI  k 


MEMUIUS    ON 

suit ;  for,  thougli  tlio  attraction  will  draw  tho  arm  aside  more 
than  it  would  otherwise  do,  yet,  as  the  accelerating  force  by 
which  the  arm  is  made  to  vibrate  is  diminished  in  the  same 
proportion,  the  square  of  the  time  of  a  vibration  will  be  in- 
creased ill  the  same  })roportion  as  the  space  by  which  the  arm 
is  drawn  aside,  and  tiiereforo  the  result  will  be  the  same  as  if 
the  case  exerted  no  attraction  ;  but,  if  the  attraction  of  the 
case  is  not  proportional  to  tiie  distance  of  tiie  arm  from  the 
middle  point,  the  ratio  in  which  the  accelerating  force  is  di- 
minished is  different  in  diiferent  parts  of  the  vibration,  and 
the  square  of  the  time  of  a  vibration  will  not  be  increased  in 
the  same  proportion  as  the  quantity  by  which  the  arm  is  drawn 
aside,  and  therefore  the  result  will  be  altered  thereby. 

On  computation,  1  find  that  the  force  by  which  the  attrac- 
tion draws  the  arm  from  the  centre  is  far  from  being  propor- 
tional to  the  distance,  but  the  whole  force  is  so  small  as  not  to 
be  worth  regarding;  for,  in  no  position  of  t!ie  arm  does  the 
attraction  of  the  case  on  the  balls  exceed  that  of  ^th  of  r«  spheric 
inch  of  water,  placed  at  the  distance  of  one  inch  from  the  cen- 
tre of  the  balls  ;  and  the  attraction  of  the  leaden  weight  equals 
that  of  10. G  spheric  feet  of  water  placed  at  8.85  inches,  or  of 
234  spheric  inches  placed  at  1  inch  distance  ;  so  that  the  at- 
traction of  the  case  on  the  balls  can  in  no  position  of  the  arm 
exceed  y^\^  of  that  of  the  weight.  The  computation  is  given 
in  the  Appendix. 

It  has  been  shown,  therefore,  that  the  force  required  to  draw 
the  arm  aside  one  division,  is  greater  than  it  would  be  if  the 
arm  had  no  weight,  in  the  ratio  of  1.0353  to  1,  and  therefore 

—  '  '  ^..j  of  the  weight  of  the  ball ;  and  moreover,  the  attraction 

of  the  weight  and  copper  rod  on  the  arm  and  both  balls  to- 
gether, exceeds  the  attraction  of  the  weight  on  the  nearest  ball. 


in  the  ratio  of  1.0199  to  1,  and  therefore  = 


1.0199 


8,739,000D 


of  the 

818N' 


weight  of  the  ball  ;  consequently  D  is  really  equal  to  - 

1.0199  N*  N**  l.Odo.J 

X        '..     '   ..,  or  ^,. ^,  instead  of „,  as  by  the  for- 


,  as  by  the  for- 


8,739,()(K)B'       10,844B*' 10,083B 

mer  computation.     It  remains  to  be  considered  how  much  this 
is  affected  by  the  position  of  the  arm. 

*  [T/iis  should  be  10,846 ;  see  note  on  pp.  90  and  91.] 

96 


THE    LAWS    UK    (iUAVlTATlON 

Suppose  the  weights  to  be  approached  to  tlio  balls  ;  let  W 
(Fig.  7)  be  the  centre  of  one  of  the  weights;  let  M  be  the  cen- 
tre of  the  nearest  ball  at  its  mean  position,  as  when  the  arm  is 
at  20  divisions  ;  let  B  be  the  point  which  it  actually  rests  at ; 
and  let  A  be  the  point  which  it  would  rest  at,  if  the  weight  was 
removed  ;  consequently,  AI5  is  the  spa(!e  by  which  it  is  drawn 
aside  by  means  of  the  attraction  ;  and  let  M/3  be  tiie  space  by 
which  it  would  be  drawn  aside,   if  the  attraction  on  it  was 


Us  to- 
t  ball, 


.0353 
|e  for- 

h  this 


W 


B 


M 


t 


to 


Pig. 


the  same  as  whcr.  it  is  at  M.     But  the  attraction  at  B  is  greater 
than  at  M,  in  the  proportion  of  \VM'^  :  WB";  and    therefore, 

AB  =  M/3  X  Y^  j^,=  M/3  X  ( 1  +  ^^h  very  nearly. 

Let  now  the  weights  be  moved  to  the  contrary  near  position, 
and  let  w  be  now  the  centre  of  the  nearest  weight,  and  h  the  point 

of  rest  of  the  centre  of  the  ball  ;  then  Ai  =  M/?xM-f  w.,r)> 

and  Bi  =  M/3x(2  +  j^,^+ ,...r  l  =  2M/3xM  +  jr|..;^    so  that  the 

whole  motion  BZ»  is  greater  than  it  would  be  if  the  attraction 
on  the  ball  was  the  same  in  all  places  as  it  is  at  M,  in  the  ratio 

of  l  +  TfTu  to  one  ;  and,  therefore,  does  not  depend   sensibly 

on  the  place  of  the  arm,  in  either  position  of  the  weights,  but 
only  on  the  quantity  of  its  motion,  by  moving  them. 

This  variation  in  the  attraction  of  the  weight,  affects  also 
the  time  of  vibration  ;  for,  suppose  the  weights  to  be  ap- 
proached to  the  balls,  let  W  be  the  centre  of  the  nearest 
weight ;  let  B  and  A  represent  the  same  things  as  before  ;  and 
let  X  be  the  centre  of  the  ball,  at  any  point  of  its  vibration  ;  let 
AB  represent  the  force  with  which  the  ball,  when  placed  at  B, 
is  drawn  towards  A  by  the  stiffness  of  the  wire  ;  then,  as  B  is 
the  point  of  rest,  the  attraction  of  the  weight  thereon  will  also 
equal  AB  ;  and,  when  the  ball  is  at  ^\  the  force  with  which  it 
is  drawn  towards  A,  by  tlie  stiffness  of  the  wire,  =  Aa;,  and  that 
with  which  it  is  drawn  in  the  contrary  direction,  by  the  attrac- 

WB' 
tion,=ABx 


W;c^ 


.^ ,  so  that  the  actual  force  by  which  it  is  drawn 
97 


towards   A  =  Aa:— 
'ZBxxAB 


MK.MUIUS    UN 
ABxWlV      .  ,,  .  „        ,„     /,      2B2:\     ,, 


VVB 


,  very  neurly.     So  that  the  actual  force  with  which 


the  ball  is  drawn  towards  the  middle  point  of  the  vibration,  is 
less  than  it  would  be  if  the  weights  were  removed,  in  the  ratio 

of  l  —  TiTTT  to  one,  and  the  square  of  the  time  of  a  vibration  is 


VVB 


3AB 


increased  in  the  ratio  of  1  to  1— Trrrr  ;  which  differs  very  little 

from  that  of    l  +  Trwr  to  1,  which  is  the  ratio  in  which  the 

motion  of  the  arm,  by  moving  the  weights  from  one  near  posi- 
tion to  the  other,  is  increased. 

The  motion  of  the  ball  answering  to  one  division  of  the  arm 

30.05* 

'    '  '  —  ;  and,  if  MBf  is  the  motion  of  the  ball  answering 

MB  30.05^/  d 


to  d  divisions  on  the  arm, 


and 


WM     :eOx  38.3x8.85     185' 

therefore,  the  time  of  vibration,  and  motion  of  the  arm,  must 

be  corrected  as  follows  : 

If  the  time  of  vibration  is  determined  by  an  experiment  in 

which  the  weights  are  in  the  near  position,  and  the  motion  of 

the  arm,  by  moving  the  weights  from  the  near  to  the  midway 

position,  is  d  divisions,  the  observed  time  must  be  diminished 

2d 
in  the  subduplicate  ratio  of  1  —  7777  ^^  1>  ^^^^  ^s>  i^  ^^^  ratio 


185 


d 


of  1  — TTjr  to  1;  but,  when  it  is  determined  by  an  experiment 

in  which  the  weights  are  in  the  midway  position,  no  correction 
must  be  applied. 

To  correct  the  motion  of  the  arm  caused  by  moving  the 
weights  from  a  near  to  tlie  midway  position,  or  the  reverse, 
observe  how  much  the  position  of  the  arm  differs  from  20 
divisions,  when  the  weights  are  in  the  near  position  ;  let  this 
be  n  divisions,  then,  if  the  arm  at  that  time  is  on  the  same  side 
of  the  division  of  30  as  the  weight,  the  observed  motion  must  be 

*  [This  number,  36.65,  here,  and  again  in  tlie  next  line,  is  eiToneously 
printed  in  Cavendis/i's  memoir  as  36.35.] 
f  [In  tlie  original  this  is  erroneously  pnnted  as  wB.] 

98 


which 

ion,  is 
3  ratio 

tioii  is 

f  little 

3h  the 
,r  posi- 

lie  arm 
waring 

- ;   and 
1,  must 

nent  in 
tion  of 
id  way 
inishcd 

|e  ratio 

iriment 
reciion 

[ig  the 
averse, 
)m  20 
let  this 
kie  side 
lust  be 

\neou8ly 


T H  K    I.  A  W  S    i)V    < ;  li  A  V  I  I'  A  'V  \  ( )  N 

"Zn 
diminished  by  tlie  part  of  tlie  wiiole ;  but,  otherwise,  it 

must  be  as  mucli  increased. 

If  the  weights  are  moved  from  one  near  position  to  the  otlier, 

and  the  motion  of  tlie  arm  is  'Zd  divisions,  the  observed  motion 

Z<1 
must  be  diminished  by  the  -—  part  of  the  whole. 

loo 

If  the  weights  are  moved  from  one  near  position  to  the  other, 

and  the  time  of  vibration  is  determined  while  the  weights  are 

in  one  of  those  positions,  tiicre  is  no  need  of  correcting  either 

the  motion  of  the  arm,  or  the  time  of  vibration.* 

Conclusion 

The  following  table  contains  the  result  of  the  experiments 


Expor. 

Mot.  weig..t 

Mot.  arm 

Do.  corr. 

Time  vib. 

Do.  corr. 

Density 

M 

m.  to  + 

14.82 

13.42 



5.5 

+  to  m. 

14.1 

13.17 

14'    55' 

— 

5.  SI 

^1 

m.  to  + 

15.87 

14.69 

— 

— 

4.88 

+  to  m. 

15.45 

14.14 

14    42 

— 

5.07 

«l 

+  to  m. 

15.22 

13.56 

14    39 

— 

5.26 

m.  to  4- 

14.5 

13.28 

14    54 

— 

5.55 

^1 

111.  to  + 

3.1 

2.95 

6'    54' 

5.36 

+  to  - 

6.18 

— 

7      1 

— 

5.29 

-  to  + 

5.92 

— 

7      3 

— 

5.58 

H 

+  to  - 

5.9 

— 

7      5 

— 

5.65 

-.  to  + 
m.  to  — 

5.98 
3.03 

2.9  ^ 

7      5 

— 

5.57 

e] 

5.53 

-  to  + 

5.9 

5.71 

7      4 
by 

5.62 

TJ 

111.  to  — 

-  to  + 

3.15 
6.1 

3.03 
5.9     ■ 

6    57 

5.29 
5.44 

8J 

m.  to  — 

3.13 

3.00 

mean 

5.34 

-   to  + 
-1-  to  - 

5.72 
6.32 

5.54^ 

6    58 

5.79 

9 

— 

5.1 

10 

+  to  - 

6.15 

— 

6    59 

— 

5.27 

11 

-f  to  - 

6.07 

— 

7      1 

— 

5.39 

12 

-   to  + 

6.09 

— 

7      3 

— 

5.42 

X3| 

-   to  + 

6.12 

— 

7      6 

— 

5.47 

+  to  - 

5.97 

— 

7      7 

— 

5.63 

u] 

-  to  -f 

6.27 

— 

7      6 

— 

5.34 

+  to  - 

6.13 

— 

7      6 

— 

5.46 

15 

-  to  + 

6.34 

— 

7      7 

— 

5.3 

16 

-  to  + 

6.1 

— 

7    16 

— 

5.75 

n| 

-  to  + 

5.78 

— 

7      2 

— 

5.68 

+  to  - 

5.64 

— 

7      3 

— 

5.85 

*  [  The  corrections  neutralize  each  other,  since  they  are  the  same  for  N*  and 
B,  whose  ratio  enters  into  the  expression  for  D.] 

99 


MK  MO  I  IIS    ON 

From  til  is  tablo  it  jippeurs,  tluit  tiiougli  tlie  experiments 
agree  pretty  well  together,  yet  the  difference  between  them, 
both  in  the  f|iijintity  of  motion  of  the  arm  and  in  the  time  of 
vibration,  is  greater  than  ean  proceed  merely  from  the  error  of 
observation.  As  to  the  difference  in  tiie  motion  of  the  arm,  it 
nniy  very  well  be  acconnted  for,  from  the  current  of  air  pro- 
duced by  the  difference  of  temperature  ;  but,  whether  tiiis  can 
account  for  the  difference  in  the  time  of  vibration,  is  doubtful. 
If  the  current  of  air  was  regular,  and  of  the  same  swiftness  in 
all  parts  of  the  vibration  of  the  ball,  1  think  it  could  not;  but, 
as  there  will  most  likely  be  nnudi  irregularity  in  the  current,  it 
may  very  likely  be  suf!icient  to  iiccount  for  the  difference. 

hy  a  mean  of  the  experiinents  made  with  the  wire  first  used, 
the  density  of  the  earth  comes  out  5.48*  times  greater  than 
that  of  water  ;  and  by  a  mean  of  those  made  with  the  latter 
wire,  it  comes  out  the  same  ;  and  the  extreme  difference  of  the 
results  of  the  ^3  observations  made  with  this  wire,  is  only  .75  ; 
so  that  the  extreme  results  do  not  differ  from  the  mean  by  more 
than  .38,  or  ^  of  the  whole,  and  therefore  the  density  should 
seem  to  be  determined  hereby,  to  great  exactness.  It,  indeed, 
may  be  objected,  that  as  the  result  appears  to  be  influenced  by 
the  current  of  air,  or  some  other  cause,  the  laws  of  which  we 
are  not  well  acquainted  with,  this  cause  may  perhaps  act  al- 
ways, or  commonly,  in  the  same  direction,  and  thereby  make  a 
considerable  error  in  the  result.  But  yet,  as  the  experiments 
were  tried  in  various  weathers,  and  with  considerable  variety 
in  the  difference  of  temperature  of  the  weights  and  air,  and 
with  the  arm  resting  at  different  distances  from  the  sides  of 
the  case,  it  seems  very  unlikely  that  this  cause  should  act  so 
uniformly  in  the  same  way,  as  to  make  the  error  of  the  mean 
result  nearly  equal  to  the  difference  between  this  and  the  ex- 


f 


*  [This  should  be  5.31.  Had  thethird  number  in  tJiecolumn  of  densities  been 
5.88,  instead  f>/4.88,  the  average  would  have  been  as  Cavendish  gave  it.  But 
Baily  (79,  p.  90)  rectilculated  tJte  densities  from  Cavendish's  data,  and  found 
4.88  to  be  correct.  Curiously  enough  Cavendish  made  the  same  error  in  deduc- 
ing the  mean  result  of  the  whole  number  of  experiments.  It  should  be  5.448,  not 
6  ^  {which  icould  be  had  by  putting  5.S8  inj)lace  r>/"4.88),  withaprobable  error 
<j/'.033.  The  mean  result  of  the  last  23  observations  is  5.48.  The  greatest  dif- 
ference of  the  single  results  from  one  another  is  .97  ;  and  tJie  extreme  result 
differs  from  the  mean  of  all  by  .57,  or  ^.^  of  tlie  whole.  For  an  account  of 
Iluttons  reflections  on  Cavendish's  ''pretty  and  amusing  little  experiment " 
see  his  paper  (45  and  54) ;  they  are  referred  to  in  this  volume  on  page  105.] 

100 


...->Li Wi*  .-'I'ltiilfclil  :>■. '.  .^^ 


es  been 

But 

I  found 

\deduc- 

18,  not 

error 

[at  dif- 

result 

mt  of 

mt,"' 

|>5.] 


THE    LAWS    OK    CiKAVITATlUN 

treme  ;  and,  thoreforo,  it  sooma  very  unlikely  that  tlio  density 
of  the  earth  sliould  dilTor  I'roin  r>.48  by  so  luueii  as  ^^^  of  tlio 
whole.* 

Another  objection,  perhaps,  may  be  made  to  these  experi- 
ments, namely,  that  it  is  uncertain  whether,  in  these  small 
distances,  the  force  of  gravity  follows  exactly  the  same  law  as 
in  greater  distances.  There  is  no  reason,  howt^ver,  to  tiiiniv 
that  any  irreguhirity  of  this  kind  takes  place,  until  the  bodies 
come  within  the  action  of  what  is  called  the  attraction  of  co- 
hesion, and  which  seems  to  extend  only  to  v(n*y  minute  dis- 
tances. With  a  view  to  see  whether  tlie  result  could  be  affected 
by  this  attraction,  I  made  the  0th,  10th,  lltli,  and  15th  exjjeri- 
ments,  in  which  the  balls  were  made  to  rest  as  close  to  the  sides 
of  the  case  as  they  (M>uld  ;  but  there  is  no  difference  to  be  de- 
pended on,  between  the  results  under  that  circumstance,  and 
when  the  balls  are  placed  in  any  other  part  of  t!ie  case. 

According  to  the  experiments  made  by  Dr.  Maskelyne,  on 
the  attraction  of  the  hill  Schehallien,  the  density  of  the  earth 
is  4|^  times  that  of  water  ;  which  differs  rather  more  from  the 
preceding  determination  than  I  should  have  expected.  lUit  I 
forbear  entering  into  any  consideration  of  which  determination 
is  most  to  be  depended  on,  till  I  have  examined  more  carefully 
how  much  the  preceding  determination  is  affected  by  irregular- 
ities whose  quantity  I  cannot  measure. 

*  [See  note  on  page  100.] 
101 


i 


APPENDIX 

ON   TITH;    attraction  of  the   MAMOdANY  (!ASE  ON  THE    BALLS 

The  lirst  tiling  is,  to  find  tlio  iittraction  of  tlie  rectangular 

;,  piano  ckiM  (Fig  8)  on  tiie 

point  a,  placed  in  tjje  line 
ac    perpendicular    to    this 
plane. 
/^       Let  ac  =  rt,  ck  =  b,  ch  — 

ic,  and  let  -5-- — ^  =  «t>',  and 
a   ■\-  X 


Fig-  8 


-s — - — ;;  =  v",   tlien    the   at- 
a   +  ;' 

traction  of  the  line  bfi  on 
a,    in    the  direction   ab,  = 

,     —. — - — , ;  and  therefore,  if 
t^    ab  X  afi' 

cb  flows,  the  fluxion  of  the 


attraction  of  the  plane  on  the  point  a,  in  the  direction  cb,  = 
bx  ^        _        ~  ^''^^      _      ~  ^^'^ 

__  -— the  variable  part  of  the  fluent  of  which  =  —  losr  (v 

4-  Vi-\-vY,  and  therefore  the  whole  attraction  =  lou:  (  — '^ — -x 

nb     \  \     ac 

vTs si  ;  "^o  that  the  attraction  of  the  plane,  in  the  direction 

cb,  is  found  readily  by  logarithms,  but  I  know  no  way  of  finding 
its  attraction  in  the  direction  ac,  except  by  an  infinite  series.* 


*  [Playfair  has  given  an  expression  in  finite  terms  for  this  attraction  on  pp. 
225-8  of  his  pajyr  in  the  Tnins.  Rov.  Soc  Edin.,  vol.  6,  1812.  pp.  187-243, 

103 


M  KM  OIKS    ON    TIIK    LAWS    OK    <;  K  A  V  I  T  A  T  I  (>  N 

Tlio  two  most  ronvcnioiit  Horios  I    know,  are  tlio  followini^  : 
Fiirt  St'i'ios.   Let  -  —  tt,  unci  lot  A=un'  wlioso  tau^.  i.s  n-,  li 

=  A— TT,  (^'  =  H-|--— ,  D— (' — .  ,  t'te.     'riuMi  llm  attniction  ill  tlio 


direction  ar 


1 


For  the  second  series,  let  A  =  iirc'.  whose  tanj;.  =    ,  IJ  =  A — , 

«■  IT 

C=:lH.i-3»  I)  =  C-;r-5,  etc.      Tlicii  tlie  attraction  =  uic.!K)" 


IJtt 


On- 


It  must  1)0  observed,  that  the  first  scries  fails  wiien  tt  is 
greater  tlian  unity,  and  the  second,  when  it  is  h;ss  ;  hut,  il'  h  is 
taken  equal  to  the  least  of  the  two  lines  rk  and  vh,  there  is  no 
case  in  which  one  or  the  other  of  them  may  not  be  used  con- 
veniently. 

By  tlie  iielp  of  these  series,  I  computed  the  following  table  : 


|log(v 
ak 

X 

action 

jmling 
'ies.* 


on  pp. 
37-243, 


.lOfi'i 

.3714 

.6145 

.6248 

.7071 

.7808 

.8676 

.9285 

9H15 

1. 

.1962 

.00001 

.3714 

.00039 

.00148 

.5145 

.00074 

.00277 

.00521 

.6248 

.00110 

.00406 

.00778 

.01183 

.7071 

.00140 

.00522 

01008 

.01525 

.02002 

.7808 

.00171 

.00637 

.01245 

.01896.02405 

03247 

.8575 

00207 

00772 

.01522 

02339.03116 

.03964 

.05057 

.9285 

.00244 

.00910 

.01810 

.02807.03778.04867 

06319 

.08119 

.9815 

.00271 

.01019 

.02084 

.03193 

.04368  .05639 

07478 

09931 

.12849 

1. 

.00284 

.01054 

.02135 

.03347 

.04560.05975 

.07978 

.10789 

.14632 

.19612 

ck 


Find  ill  this  table,  with  the  argument  ~,  at  top,  and  the  ar- 

(tic 

gument  -^  in  the  left-hand  column,  the  corresponding  logarithm ; 


entitled  "  Of  the  Solids  of  Gredtent  Attraction,  or  those  which,  among  all  the 
Solids  that  have  certain  Properties,  attract  with  the  greatest  Force  in  a  given 
Direction.'^'] 
*  [In  the  la  fit  term  of  the  series  Ute  cotfjicicni  D  was  omitted  in  the  original.] 

103 


i 


M KM  OIKS    ON 


.JKi' 


m 


1'  w^^ 


tlioji  JuM  toorotlior  tliis  lojjjiritlini,  tlio  lofjiiritlirii  of     , ,  suid   tlio 


rfj 


nk' 


i<)*;uritlim  of     .  ;  tlic  sum  is  tlio  loguritlirn  of  the  uttniction. 

To  ('onipiite  from   licm'.u  Llio  sittnictiou   of  tlu;  ciiso  ofi  tlio 
^  E  ball,  lot  thobox  l)(!BA(Fi^'. 

1),  in  which  th(!  ball  pluyK,  bo 
tiividod  into  two  piirts,  by  a 
vortical  sootioii,  porpendio- 
ular  to  tho  length  of  tho 
caHo,  and  passinji^  through 
the  centre  of  the  ball  ;  and, 
in  Kig.  '.),  Kit  the  parallel- 
opij)e(l  \\\\)Knh<lc  bo  one  of 
these  [)arts,  ABDK  being  tho 
above-men tionod  vertical  sec- 
tion ;  let  X  bo  tho  centre  of 
tho  ball,  and  draw  t!ie  par- 
allelogram i^npnilx  parallel  to 
IVvr/I),  and  xfirp  parallel  to 
fiWbn,  and  bisect  /5^  in  c. 
Now,  the  dimensions  of  the 
box,  on  tho  inside,  arc  Bd=:1.75  ;  Hi)  =  ;J.(5 ;  H/3=1.7r);  and 
/5A  — 5 ;  whem^c  I  find  that,  if  xc  and  /3./'  are  taken  as  \\\  tho  two 
upper  linos  of  tho  following  table,  the  attractions  of  the  differ- 
ent parts  are  as  set  down  below. 


B 


1   \ 

n 

\ 

1   A 

X 

c 

i\ 

I 

1 

K 

p 

\ 

1   n 

1 
1 

\ 

\ 

\ 

o\^ 

d' 

m 


Fig.  9 


Excess  of  attract  ion 


Slim  of  these 
Excess  of  attraction 


xc 

ftx 

of  Ddi'd  above  Bhrg 

mdr p  above  nb  r  p 

mesp  al)ove  nasp 


of  BA;j/3  al)Ove  DdmS 
Aanfi  above  EeviS 

Whole  attraction  of  the  inside  surface  of  the 
half  box. 


.75 
1.05 
.2374 
.2374 
.3705 

.8453 
.5007 
.4677 

.1231 

.5 
1.3 
.1614 
.1614 
.2516 

"5744 
.3271 
.3079 

.0606 

.25 
1.55 
.0813 
.0813 

.2897 
.1606 
.1525 


.0234 


It  appears,  therefore,  that  tho  attraction  of  tho  box  on  x  in- 
creases faster  than  in  proportion  to  the  distance  xc. 

The  specific  gravity  of  tho  wood  used  in  this  case  is  .61,  and 
its  thickness  is  f  of  an  inch  ;  and  therefore,  if  the  attraction 
of  the  outside  surface  of  the  box  was  the  same  as  that  of  the 

104 


THK    LAWS    OK    (iUAV  11  A  T  I  ( )  N 

inside,  tho  whole  attniotioii  of  the  hox  on  the  ball,  when  rr 
=  .75,  would  he  equal  to  2  x.r>31  x.'il  x  I  cnhic!  inches,  or  .'H)\ 
ephericj  inches  of  water,  placed  at  the  distance  of  one  inch  from 
the  centre  of  the  ball.  In  reality  it  vm\  never  bo  so  ^reat  as 
this,  as  the  attraction  of  the  outside  siirfacui  is  rather  less  than 
that  of  the  in-^ide  ;  ami,  moreover,  the  <listan<H»  of  ./•  from  r  can 
never  he  quite  so  great  as  .To  of  an  inch,  as  the  greatest  motion 
of  the  arm  is  only  l^  inch. 


.25 
.55 

.0813 
.0813 
,1271^ 

:2897 
,1606 
■  1525 

.0234 


X  in- 

L  and 

Iction 
If  the 


Much  has  been  written  concerning  the  Cavendish  experi- 
ment ;  the  following  references  may  be  consulted  to  advantage. 

( I  ilbert  (40),  in  1 71M),  translated  the  greater  part  of  ( 'avendish's 
paper  into  German  for  his  Anmilcn,  adding  many  explanatory 
notes.  A  few  years  later  Jirandes  (4"^)  gave  a  frc^sh  mathemati- 
cal analysis  of  the  experiment,  incduding  the  equations  for  the 
time  of  swing  of  the  torsion  pendulum  in  the  experiment  pro- 
posed by  Muncke  (see  below).  In  181."),  the  original  j)aper  of 
Cavendish  was  translated  entire  into  French  by  M.  Chomprc 
(50). 

In  1821,  Hutton  (54)  recalculated  the  results  of  the  experi- 
ment after  Cavendish's  own  formuhe,  ami  found,  as  he  thought, 
a  "copious  list  of  errata,  some  of  which  are  large  or  import- 
ant." Tho  mean  of  the  first  0  experiments  so  corrected  is  5.19, 
and  of  the  other  23  is  5.43  ;  the  mean  of  these  two  means  is 
5.31,  which  Hutton  takes  as  the  correct  result  given  by  the 
Cavendish  experiment.  Baily  states,  however  (79,  pp.  92-9()), 
that  Hutton  himself  had  fallen  into  error,  and  that  the  com- 
putations of  Cavendish  are  correct  except  in  the  one  detail  re- 
ferred to  on  page  100  of  this  volume.  Haily  gives  a  very  care- 
ful criticism  of  the  experiment  on  pp.  88-91  of  his  memoir.  He 
remarks  that  *' Cavendish's  object,  in  drawing  up  his  memoir, 
appears  to  have  been  more  for  the  purpose  of  exhibiting  a 
specimen  of  what  he  considered  to  be  an  excellent  method  of 
determining  tills  important  inquiry,  than  of  deducing  a  result 
that  should  lay  claim  to  the  full  confidence  of  the  scientific 
world."  Baily  points  out  that  the  time  was  not  determined 
with  due  accuracy  ;  that  the  experiments  were  not  arranged  in 
groups,  in  order  to  eliminate  the  error  arising  from  the  march 
of  the  resting  point;  and  that  the  distance  between  tiie  weight, 

105 


MKMolltS    ON 


I 


I 


1111(1  tlic  hull  WHH  jiHHumod  «MniHtut)t.  Wo  hIiuII  SCO  liitor  from 
tlio  u<;(;oiiiit.s  of  tlii>  iiivoMti;;iit ions  of  JNtioli,  Kiiily,  ('oriiii  itiul 
hiiillc,  imd  hoyw  how  tho  ornns  in  (.'iivculish's  ox|)(U'iriioiit  liiivo 
hcM'M  uvoiih'il. 

MiiiKtko  (<l|,  vol.  .*{,  pp.  040-70)  Ims  ^ivoii  iin  jiooount  of  tlio 
oxpcrinuMit  iind  an  lulrninihlo  critioisin  of  it,  and  cornparos  the 
result  with  that  ohtaiiiiMi  hy  MaskclyiKt  and  Mutton.  Il(!  pro- 
positi another  nicthod  of  nsin^  the  torsion  halani^t^  to  Iind  tho 
in(!an  density  of  tlu^  earth  ;  he  wonld  Iind  the  tini(;of  vihratioii 
with  th(^  masses  lirst  in  the  line  of  the  ))allH  and  then  in  a 
line  at  ri^lit  anj^les  to  that  direetion.  'I'lierc;  would  ho  no  de- 
llection  to  ho  !neaHiir«Ml.  We  Inivo  seen  ahove  that  Brandos 
pive  the  theory  of  this  experiment.  Investigations  of  this 
nature  have  heen  made  hy  liei(di  (81)),  Kotvos  (M.)'-i)  and 
Hraun  (H)JJ);  for  accounts  of  whicdi  see  the  latter  part  of  this 
volume. 

A  useful  resume  of  Cavendish's  paper  was  j^iven  hy  Sehmidt 
((14,  vol.  '4,  pp.  4H1-7).  lie  formed  anew  equations  from  which 
to  derive  the  value  of  the  density  of  tho  earth,  and  found  5.52 
usinj(  Cavendish's  data. 

Anotluir  mathematical  investigation  of  the  dynamical  proh- 
lem  underlying  the  Caveiulish  experiment  was  made  hy  Mena- 
brea  (71,  7'i  and  713),  in  l.S4(>.  It  is  a  very  elahorato  analysis  of 
the  whole  prohlom.  He  examines  the  effect  of  the  resistance 
of  the  air  on  the  time  of  vihration,  and  also  shews  how  to  find 
the  fuass  of  the  earth  supposing  tliat  it  is  composed  of  spheroidal 
layers  of  variable  density.  In  Baily's  memoir  (7!>)  is  another 
elaborate  analysis,  hy  Airy,  of  the  matheunitical  theory  of  the 
investigation.  It  treats  especially  of  Baily's  modificatioii  of 
the  Cavendish  experiment  (reproduced  in  Routh's  Jliffid  Dy- 
namics 1882,  pt.  1,  pp.  350-304). 

An  elementary  treatment  of  the  problem  involved  is  given  by 
Cosseiin  (127),  and  from  the  formula  he  arrives  at  he  derives 
the  value  of  the  mean  density  of  the  earth  as  given  by  Caven- 
dish's experiment,  and  gets  5.00.  A  similarly  elementary  treat- 
ment by  liabinet  (132)  gives  5.5. 

An  excellent  account  of  Cavendish's  work  is  given  by  Zanotti- 
Bianco  (148|)  and  by  Poynting  (185,  pp.  40-8) ;  in  the  latter  is 
to  be  found  a  diagram  showing  the  closeness  of  Cavendish's 
separate  results  to  the  mean. 

106 


;n  by 
Irives 
[von- 
[•etit- 


Tll  K    LAWS    OK    (JKAV  I  T  A  I' h  >  N 

IIkNUY    Cw  TA'DlSIf,  HOU  of    LfU'd   ( 'llJlllt'S  ( IjlViMulisIl    uiid    a 

lu'plu'w  of  tlic  third  I)iikr  of  l>t'Voii«liir«'.  wha  horn  ut  Nico  in 
lT>n  litiil  (liotl  ut  Ijondoii  in  1K|(».  |[e  stiidiod  ut  ('iiinl)rid<{(>, 
and  luuioniin;;  poMsrHscd,  l>y  tho  di'iith  of  an  uncit',  of  >i  hir^o 
fortiint^  hu  di'votcd  his  iifu  unostcntutiously  to  privntr  KvMcntilic 
studies.  Hcsid(s  i\\v.  inv(>sti.L,^ition  on  <^n'avitalion:il  attraction 
h(>r(>  n>print(>(l,  he  is  rrrnarkahh'  for  liis  rcscandics  in  th«>  fudd 
of  (duMnistry,  and  has  hocn  calitMl  tlio  '*  Newton  "  of  that  suh- 
je(;t.  lie  worked  on  the  eonstitnents  of  the  atmosphere  ami 
on  hydro^^en  ;  he  made  tho  first  syntiu\sis  of  water,  liy  Inirnin^ 
iiydroifen  in  air.  and  found  thti  density  of  liydro^en  to  lio  ,', 
(instead  of  ^4)  of  tliat  of  air.  lie  det(M*min(M|  the  ratio  of  de- 
pido<fistieated  to  phloj^istieated  air  to  he  about  as  1  :  4.  Caven- 
dish also  made  numy  rc^searelies  of  «;reat  importamio  in  th»!  sub- 
ject of  (d(!etri(,'ity  ;  these  iiave  been  coibHjtcd  and  edited  by 
Clerk-Maxwell,  i'erhaps  the  most  important  of  his  electricMil 
investigations  is  that  whi(di  proved  that  (dectrostatic  attrac^tion 
takes  ])lace  according  to  the  law  of  the  inverse  s«|uare  of  the 
distance,  lie  is  also  the  author  of  sev(M'al  paptu's  on  astromdui- 
oal  questions.  Most  of  his  writings  are  to  be  found  in  the 
Philosophical  Transactions  of  the  period. 

107 


lotti- 

jr  is 

lish's 


illl 


JSgjBr     -^r«jv^T-,;:,^i< 


HISTORICAL  ACCOUNT  OF  THE  EXPERT 

MENTS  MADE  SINCE  THE  TIME 

OF  CAVENDISH 


fv|^  ■■^?^  ((«ii  ii!-Ti«f  ^pkivn^'ijwfljfj'ii"!  ■  M  i|uvm  imm 


m 


HISTORICAL  ACCOUNT  OF  THE  EXPERI- 
MENTS MADE  SINCE  THE  TIME 
OF  CAVENDISH 


Carlini.  Ill  1821,  Cfirliiii,  director  of  tlie  Hrera  ohservtitory 
iit  Milan,  made  a  series  of  experiments  at  tlie  Hospice  on  Mt. 
Cenib  in  the  Alps,  to  determine  tlie  length  of  the  seconds-pend- 
ulum (55).  He  was  led  to  do  so  from  considering  that  tiio 
Alps  offerad  a  favourable  situation  for  a  determination  of  the 
mean  density  of  the  earth,  and  that  no  pendulum  experiments 
had  been  made  there  since  the  publication  of  the  fictitious 
ones  of  Coultaud  and  Mercier  (see  p.  47).  Carl  in  i  compared 
the  time  of  vibration  of  a  simple  pendulum,  made  after  the 
general  style  of  Borda's,  with  that  of  a  standard  clock  whose 
rate  was  noted  daily.  The  height  of  the  observing  station  was 
194:3  metres  above  sea-level,  in  latitude  45°  14'  10".  The  cor- 
rected length  of  the  seconds -pendulum  reduced  to  sea -level 
was  found  to  be  993.708  mm.  Matthieu  and  Biot  had  found 
the  length  of  the  "decimal"  seconds-pendulum  at  Bordeaux, 
in  lat.  44°  50'  25",  to  be  741.G151  mm.  The  calculated  length 
at  Mt.  Cenis  would  be  741.0421  mm. ;  or,  for  the  *'  sexagesimal  " 
seconds -pendulum  (100  000  decimal  seconds  =  80  400  sexages- 
imal seconds)  993.498  mm.  The  difference  between  this  length 
and  the  observed  length  is  .210  mm.,  which  represents  the  at- 
traction of  the  mountain  on  the  pendulum. 

The  mountain  Is  composed  of  schist,  marble  and  gypsum,  of 
specific  gravities  2.81,  2. 80  and  2.32  respectively.  Carlini  took 
the  average  of  all  three,  2.00,  as  the  mean  density  of  the  hill. 
Assuming  tliat  the  hill  was  a  segment  of  a  sphere  1  geographical 
mile  in  height  and  had  a  base  of  11  miles  in  diameter,  the  at- 
traction was  calculated  to  be  5.0203,  where  B  is  the  specific 
gravity  of  the  hill.     With  the  same  units  the  attraction  of  the 

111 


MEMOIRS    ON 


I 


eartli  is  14  IK)4A,  where  A  is  the  mean  density  of  the  earth.* 
wi  ^-^^^^^^         -^10  ,  ^     ,  on 

We  cannot  place  very  great  confulence  in  this  result,  not  only 
on  account  of  the  fact  that  the  oxtreme  value  of  the  length  of 
the  seconds-pendulum  varies  from  the  mean  by  .032  mm.  and 
only  13  determinations  were  made  ;  but  especially  because  the 
size  and  density  to  be  assigned  to  the  mountain  are  largely  a 
matter  of  conjecture. 

A  resume  of  Carlini's  paper  was  given  by  Saigey  (56),  and 
by  Schell  (135).  Excellent  accounts  of  the  experiment,  with 
criticisms  of  it,  have  been  given  by  Zanotti-Bianco  (148ii^,  P^*  '^f 
pp.  130-45),  Poynting  (185,  pp.  22-i)  and  Fresdorf  (186^,  pp. 
8-11). 

Sabine  (58  and  82,  notes,  p.  47)  remarks  that  Biot  and  Car- 
lini  had  not  properly  reduced  to  vacuo  the  observed  pendulum 
lengths,  and  states  that  the  corrected  length  of  ^,he  seconds- 
pendulum  on  Mt.  Cenis  is  39.0992  in.  From  the  observations 
made  in  the  Formentera-Dunkirk  survey  he  finds  by  interpo- 
lation a  pendulum  length  of  39.1154  in.  for  the  latitude  of  the 
Hospice.  The  difference  between  the  observed  and  calculated 
lengths  is  .01G2  in.  The  difference  calculated  from  the  inverse- 
square  law  is  .0238  in.  With  Carlini's  data  and  equations  he 
derives  4.77  for  the  value  of  A. 

Schmidt  (G4,  vol.  2,  p.  480)  gives  a  concise  account  of  the 
theory  of  the  experiment,  and  remarks  that  Carlini  made  an 
error  in  determining  the  attraction  of  a  spherical  segment. 
Making  the  necessary  correction  he  finds  A  =  4.837,  a  result 
not  far  from  that  of  the  Schehallien  experiment. 

In  1840,  Giulio  (70)  also  gave  the  true  expression  for  the  at- 
traction of  a  spherical  segment,  and  noted  that  several  other 
corrections  must  be  made  in  Carlini's  calculations  ;  the  height 
of  the  segment  is  1.05  miles,  instead  of  1  mile  ;  the  length  of 
the  pendulum  as  determiued  by  Biot  must  be  corrected  for  an 
error  in  the  rule  used  to  find  the  length,  and  for  the  altitude  of 
Bordeaux.  Moreover,  the  red  uctions  to  vacuo  had  not  been  made 
properly  in  either  case.  When  all  these  corrections  had  been 
applied  to  Carlini's  results,  Giulio  found  the  value  of  A  to  be  4.95. 

*  This  sigoificatioD  of  A  will  be  retained  throughout  the  rest  of  the 
.volume. 

112 


THE    LAWS    OF    (J  II  A  V  I  T  A  T  I  ()  N 

Saigey  (74,  p.  155)  niiikos  tlio  observed  pendiiliini  length 
corrected  to  vacuo  !)9;J.75r)  mm.,  a!id  tlio  calculated  length 
903.017  mm.     With  thesr'  uumbers  the  value  of  A  becomes  0.15. 

Zanotti- Bianco  (148|^,  pt.  2,  p.  130)  mentions  that  Knopf 
(149J)  has  compared  the  value  of  gravity  iis  observed  by  C'ar- 
lini  on  the  top  of  the  tnountaiu  with  the  value  calculated  for 
the  same  place  from  observations  made  on  the  same  parallel  of 
latitude,  and  found  for  A,  5.08. 


.f  the 

ie  an 

lent. 

•esult 

at- 
)ther 
jight 
th  of 
)r  an 
le  of 

»ade 
Ibeen 

.95. 


If  the 


Airy,  Wiiewell  and  SnRRPSFiA>fKs  at  Dolooatii  Mink. 
In  1820,  Drobisch,  in  an  appendix  to  si  piimphlet  on  the  figure 
of  the  moon  (57),  suggested  that  experiuients  be  made  on  tlu^ 
change  in  the  period  of  a  pendulum  when  carried  from  the  sur- 
face of  the  earth  to  the  bottom  of  a  mine;  he  gave  the  theory 
of  the  experiments  and  calculated  the  change  resulting  from 
certain  hypotheses.  It  is  interesting  to  recall  the  fact  that 
Bacon  proposed  the  same  investigation  two  centuries  earlier. 
(See  p.  I.) 

At  the  very  same  time,  unknown  to  Drobisch,  experiments 
of  this  nature  were  being  tried  in  England  by  Airy  and  Whew- 
ell  at  the  copper  mine  of  Dolcoath  in  Cornwall.  Their  meth- 
od was  to  swing  one  invariable  pendulum  at  the  mouth  of 
the  pit  and  compare  its  rate,  by  Kater's  method  of  coincidences, 
with  that  of  a  standard  clock,  and  at  the  same  time  perform 
the  same  operation  upon  anoilier  pendulum  and  another  clock 
at  a  depth  of  1220  ft.  in  the  mine.  The  pendulums  were  then 
exchanged  and  the  operations  repeated.  The  greatest  difficulty 
experienced  was  that  of  comparing  the  rates  of  the  two  clocks. 
The  first  series  of  experiments  was  abruptly  stopped  on  account 
of  the  damage  received  from  fire  by  the  lower  pendulum.  A 
short  account  of  the  method  was  published  in  1827  (00),  and 
Drobisch  translated  it  for  Poggendorlf's  Annalen  (59),  wherein 
he  gives  also  a  more  complete  account  of  the  tiieory  and  an  ap- 
plication of  hjs  equations  to  Airy's  observations.  Assuming 
the  mean  density  of  the  surface  layer  of  the  earth  to  be  2.587, 
the  experiments  gave  about  20  for  the  value  of  A.  Drobisch 
contends  that  the  surface  density  should  be  taken  to  be  1.52, 
considering  how  large  an  amount  of  the  surface  layer  is  water. 

Two  years  later  Airy  and  Whewell,  assisted  by  Mr.  Sheep- 
shanks and  others,  attempted  to  repeat  the  experiments  ;  but 
after  overcoming  various  anomalies  in  the  motions  of  the  pend- 
H  113 


mi:  MO  I  lis  ON 

uhunH,  ilio  observations  \V(Mo  stopped  by  a  fall  of  rock  in  the 
mine.  The  value  of  A  foinul  from  this  series  was  about  0.  A 
full  account  of  the  exi)eriments  was  print(Ml  j)rivately  ((>:i)  in 
IHJiH,  and  Drobisch  translated  the  pamphlet  for  the  Annalen 
(03). 


)■ 


Reich's  First  Fxpruiment.  In  1838,  F.  Tieieh.  Professor 
of  Physi(!s  ill  the  Herijfakademie  at  Freiberc^,  ])iil)lishe(l  in  book 
form  (07)  the  account  of  a  series  of  experiments  carried  on  by 
him  since  1835  to  find,  after  the  method  of  Cavendish,  the 
mean  density  of  the  earth.  The  adoption  of  the  mirror  and 
scale  method  of  measuring  deflections  seemed  to  iiim  to  prom- 
ise a  !n(!Jins  of  overcoming  many  of  the  ditticulties  against  whicdi 
('avondish  had  contended.  The  final  observations  were  made 
in  the  year  1837. 

In  order  to  avoid  the  effects  due  to  irregularities  of  tempera- 
ture, the  api)aratus  was  set  up  in  a  cellar  room  wiiich  was  care- 
fully c1os(m1  up,  and  the  observations  made  through  a  hole  in 
the  door.  The  arm  of  the  balance  was  '.i.Oll)  m.  long,  and  its 
moment  of  inertia  was  found  after  the  manner  used  by  (Jauss 
for  a  magnet.  Tiie  average  weight  of  each  of  the  balls  was 
484.213  gr.,  and  their  distance  below  the  arm  was  77  cm.  They 
were  composod  of  an  alloy  of  about  1)0  parts  tin,  10  parts  bis- 
muth, and  a  little  lead.  The  attracting  masses  were  of  lead 
45  kg.  in  weight  and  about  20  cm.  in  diameter,  and  hence 
much  smaller  than  those  used  by  Cavendish.  They  were  sus- 
pended from  pulleys  running  on  rails  parallel  to  the  arm  of  the 
balance,  and  could  be  quickly  moved  from  the  null  to  the  at- 
tracting positions.  Only  one  mass  was  in  the  attracting  posi- 
tion at  a  time,  on  account  of  the  fsict  that  in  every  one  of  the 
four  attracting  positions  the  distance  from  the  mass  to  the  ball 
was  slightly  different ;  whereas  Cavendish  used  both  masses  at 
once.  The  distance  from  mass  to  ball  was  measured  at  each 
observation  by  means  of  a  telescope  moving  along  a  horizontal 
scale,  and  not  once  for  all  as  was  done  by  Cavendish. 

After  the  suspended  system  was  set  up,  Reich  found  a  con- 
tinual changing  of  the  zero-point,  which  often  lasted  for  G 
months.  In  his  final  observations  this  was  not  noticeable  be- 
cause of  the  length  of  time,  1^  years,  which  intervened  between 
the  initial  and  final  experiments.  Accordingly  the  second 
means  of  the  elongations  were  found  by  him  to  be  more  con- 

114 


Til  K    LAWS    (H-    <i  i;.\  V  IT.\Th»N 

stunt  tlian  Ciivcndisli  found  (lu'in.  'I'lu'  Tollowinj^'  tuhlu  will 
show  this,  iind  Jilso  illustnilo  Kuicli's  nictlioil  of  tiniliii^  tlii' 
timo  of  vibisition  : 


hence 


pos\- 
of  tiro 
Hi  ball 
3ses  at 

each 
Izontal 


con- 

for  G 

)le  be- 

jtween 

lecond 

con- 


Tiinoiir 

IHlHSilKO 

Kxlrcmos 

IhI  meiiii 

'2i\  iiicaii 

III  74  5 

Hi    ' 

5.5 

05.3  . 

c  •  ■ 

7(5.00 

iv  37  yo  H 

IV'   37 

yO'.H 

50.8 

( 

7a.  H5 

74.035 

^4    31     3 

34 

30   .3 

00.9 

f 

75.00 

74.K75 

41     33  .4 

41 

13  .4 

60.0 

87.6^  '" 

74.35 

75.075 

47    59  .3 

48 

10  .8 

Av(M-ii,i;(!  or  M  inciin  74.0583 
Time  of  vibniiioii  (it'U'niiiiicil  froni  |)ii.s.«iip'  of  74.5 

IV"   41'   33".4-  IV"   37    ;'{G.8-i;}    'ir;  .i\ )  ,.,,    ....^ 

47  50  .3-  U    31  .3=13    38  .0  j  "^'''-  ^^    *^  '^ 
Time  of  vibration  (letcrmincd  from  piissa^c  of  75.5 

IV"  41'    13".4-IV"  37'    30'.H^13'    i\   A\ }  ,„,...   ,..,    ,.„  .. 

48  10.8-  34    30.3  =  13    41  .0  [  *'^'''-  ^"^    '*^    " 

Wy  interpolation  the  time  of  a  double  vibration  across  74.!)r).s;j 
is  II)' 41". 70S.  This  differs  somewhat  from  Cavendish's  method, 
as  will  be  seen  by  a  reference  to  page  (15.  lieich  considered  it 
a  more  accurate  method  tinui  that  of  ('avendish,  and  remarks 
that  when  he  applied  the  hitter's  method  to  the  above  observa- 
tions he  got  results  not  very  ditrerent,  but  on  ap[)lying  his  own 
method  to  Cavendish's  observations  lie  got  somewluit  dit!erent 
results.  Heidi's  method  was  adopted  by  Baily  (70,  pp.  44  and 
47)  in  his  experiments. 

When  in  the  above  way  at  least  three  passages  across  the  two 
median  points  bad  been  observed,  lieich  waited  until  the  arm 
was  in  its  next  extreme  position  and  seemingly  at  rest  and  then 
rapidly  moved  the  attracting  mass  to  its  new  position.  It  was 
always  so  movoil  as  to  increase  the  swing.  He  assumed  that 
the  motion  was  instantaneous,  and  used  the  last  extreme  of  the 
one  series  as  the  first  of  the  next.  Baily  (79,  p.  4G)  followed 
him  in  this  departure  from  the  procedure  of  Cavendish.  Cornu 
arul  Bailie  (14:i)  have  pointed  out  that  this  method  leads  to 
error,  and  we  shall  refer  to  the  matter  again  when  describing 
the  results  obtained  by  Baily  (page  11(5).  From  each  such  set 
of  4  extremes  the  restiug-point  and  the  time  of  vibration  arc 

115 


MEMOIRS    ON 

found.  From  )l  sucli  sets  tlio  dcviution  could  bo  found,  uiid 
the  mciui  density  of  tlie  eurtli  culeu'.ated.  iii>i(di  did  not,  how- 
cvor,  procood  in  tluit  way,  but  deduced  one  value  of  A  from  all 
the  observations  of  each  day  ;  that  is,  he  took  the  average  of 
all  the  deviations  of  that  dav  for  the  final  mean  deviation,  and 
the  average  of  all  the  times  of  vibration  for  the  llnal  nu'an  time 
of  vibration,  Jind  from  these  deduced  one  value  for  the  mean 
density  of  the  earth. 

I  na})|)lying  corrections  to  the  equations  derived  from  a  simpli- 
fied form  of  the  theory  of  the  experiment,  Keich  followed  Cav- 
endish exactly.  57  observations  were  made,  from  which  14  de- 
terminations of  the  value  of  A  were  deduced.  The  mean  of  all, 
when  corrected  for  the  centrifugal  force,  was  5.44 ±.0233,  a  re- 
sult almost  coinciding  with  Cavendisli's.  Reich  admits  at  the 
end  of  his  paper  that  there  were  certain  anomalies  in  the 
motioft  of  the  beam  which  he  could  not  account  for. 

A  second  series  of  0  observations  with  iron  masses  30  kg.  in 
iveight  and  20  cm.  in  diameter  gave  for  A,  5.4522,  which  proves 
that  no  disturbance  could  have  arisen  from  magnetic  action. 

Valuable  concise  accounts  of  Reich's  experiment  are  given 
by  Beaumont  (00),  Baily  (08,  09  and  70,  pp.  00-8),  Schell 
(135),  Poynting  (185,  pp.  48-50)  and  Fresdorf  (180^,  pp.  20-2). 

Baily.  While  Reich  was  making  the  investigations  just  re- 
ferred to,  a  very  comprcliensive  and  elaborate  series  of  experi- 
ments upon  almost  the  same  plan  was  being  carried  on  by  the 
English  astronomer  F.  Baily.  These  experiments  were  under- 
taken at  the  instance  of  the  Royal  Astronomical  Society,  and 
in  aid  of  them  a  grant  of  £500  was  made  by  the  British  govern- 
ment. The  results  were  published  in  1843  (79).  They  were 
carried  on  in  one  of  the  rooms  of  Baily's  residence,  a  one-story 
house  standing  detached  in  a  large  garden.  The  apparatus  was 
almost  the  counterpart  of  that  of  Cavendish,  except  that  the 
balls  were  not  suspended  from  the  balance  arm,  but  were 
screwed  directly  on  to  its  ends.  The  balance  and  its  mahogany 
case  were,  moreover,  suspended  from  the  ceiling,  and  the  at- 
tracting masses  rested  on  the  ends  of  a  plank  movable  on  a 
pillar  rising  from  the  floor.  As  a  protection  against  changes 
of  temperature  this  apparatus  was  then  surrounded  by  a  wooden 
enclosure.  The  masses  were  of  lead  rather  more  that  12  in.  in 
diameter,  weighing  380.409  lbs.  each.     Torsion  rods  of  deal  and 

116 


TJIE    LAWS    OK    (JliAVITATluN 


on  a 
|anges 

)oden 
liii.  in 
111  and 


of  brass,  each  about  77  in.  long,  wore  omployed,  and  their 
motion  was  observed  by  the  mirror  and  seaU>  nuitliod.  lialKs  of 
(HtTerent  materials  and  of  various  diameters  weic  experimented 
upon  :  viz.,  1.5  in.  platinum,  "i  in.  lead,  2  in.  /iiie,  2  in.  glass, 
2  in.  ivory,  2.5  in.  lead,  and  '^.h  in.  hollow  brass.  'IMie  mode 
of  suspension  was  varied  greatly,  both  single  and  double  sus- 
pension wires  being  used,  and  the  material  and  distance  apart 
of  tho  bifilar  wires  being  frerjuently  changed.  The  lengtli  of 
the  suspending  wires  was  ordinarily  al)out  00  in.,  and  tho  time 
of  vibration  varied  from  about  100  to  580  seconds. 

The  experiments  were  begun  in  Oct.  IS;J8,  and  carried  on 
for  18  months,  until  about  1300  observations  had  been  made  ; 
when,  on  account  of  tho  great  discordance  of  the  results,  a 
stop  was  nuide.  Prof.  Forbes  suggested  that  these  anomalies 
might  arise  from  radiation  of  heat,  and  advised  the  use  of  gilt 
balls  and  a  gilt  case.  These  changes  were  made,  and  the  tor- 
sion box  also  lined  with  thick  flannel.  They  ttirned  out  to  be 
decided  improvements,  although  sotne  anomalies  still  existed, 
and  it  is  evident  that  the  choice  of  a  place  for  setting  up  the 
apparjitus  was  not  a  good  one. 

Baily  adopted  the  method  of  Ueich  for  reducing  the  time  re- 
quired to  make  the  number  of  turning-points  recjuisite  for  cal- 
culating the  deviation  and  period  ;  that  is,  the  masses-  were 
moved  quickly  from  one  near  position  to  the  other,  and  the  last 
turning-point  of  one  series  served  for  the  tirst  of  the  next. 
Three  new  turning-points  were  observed  at  each  position  of  the 
masses,  and  each  group  of  4  was  called  an  **  experiment." 
2153  such  experiments  were  made  during  the  years  1841-2. 
The  time  of  vibration  was  found  for  each  experiment  after  the 
method  adopted  by  Reich.  In  deducing  the  mean  density  of 
the  earth  from  the  observations  Baily  proceeded  quite  differ- 
ently from  Reich.  There  was  always  a  slow  motion  of  the  zero- 
point,  and  Baily,  in  order  to  fake  account  of  this,  combined 
the  deflections  and  periods  in  threes.  The  difference  between 
the  deflection^'of  the  2d  experiment  and  the  average  of  the  1st 
and  3d  is  twice  the  mean  deviation.  The  average  of  the  period 
of  the  2d  experiment  with  the  average  of  the  1st  and  3d  is  the 
mean  period.  From  the  mean  deviation  and  mean  period  so 
found  a  value  of  A  is  deduced.  Another  was  then  found  from 
comparing  the  3d  experiment  with  the  2d  and  4th,  and  so  on. 
The  mean  of  all  the  experiments  gave  forA,  5.G747zt.0038.   Some 

117 


T 


■!»  ^am*  «.i*i»if.  f'^t'i 


I 


M  KM  o  I  lis    ON 

of  tli(»  oxpcrinuMits  woro  miulc  with  the  brass  rod  uloiio,  without 
any  halls,  \\h<!  mean  result  for  \vlii(;li  was  .'>.(i(i(l(;d=.<K>:jS. 

Till'  Miatlicmatical  analysis  of  th(»  prolijcni  was  ;;ivt'ii  by 
Airy,  aiul  is  iii('ori)orati'(l  in  Haily's  paper  (ID,  pp.  !i'.>-lll);  it 
is  also  to  bo  found  iu  lioutli's  Uiijid  hytntniirs,  lS8:i,  pt.  1, 
]>p.  :jr»!)-(l4. 

liaily  published  a  oondens(Ml  a(!(!ount,  of  his  work  in  several 
journals  (?'),  7'»,  77,  7.S  iind  HO).  A  cari^ful  diseussiou  of  it  is 
given  by  ScludI  (i:{"»)  ami  by  l*oyntin<;  (is:»,  pp.  r>ti-7). 

In  lS4v*,  Sai<,'ey  (M)  wrote  a  full  aeeonnt  of  all  the  experi- 
in(Mits  nuide  befori!  that  date  ;  \w  ju^ives  his  reasons  for  consider- 
ing th(^  pcMidnluin  nu'tluul  of  finding  A  the  htast  aecurati;,  the 
mountain  method  somewhat  better,  and  the  torsion  method  thu 
best,  lie  limls  great  fault  witli  the  work  of  liaily,  aiul  eon- 
Hidora  that  iiis  results  are  not  so  wor^hv  of  (M)nlide!H;e  as  thoso 
of  Cavendish.  Saigi^y  eontiMuls  that  the  anomalicss  observed 
by  ('avendish,  IJeitdi,  aiul  i^tily  cannot  be  accounted  for  by 
radiation  of  heat,  as  Korbes  suggested,  because  the  bahmco 
swings  in  an  enclosure  all  points  of  which  are  at  the  same  tem- 
peratur(f  (thus  begging  the  (juestioii);  he  (lonlldently  renuirks 
that  these  anomalies  are  caus(Ml  by  the  passage  of  air  into  or 
out  of  the  case  astluf  barometric;  pressure  (diang(\s.  The  values 
of  A  found  by  Haily  increased  from  T).!)!  to  5.77  as  the  density 
of  the  balls  used  changed  from  'Z\A)  to  1.'.)  respectividy;  Saigey 
thinks  that  this  must  arise  from  an  error  in  calculating  the 
monuuit  of  inertia  of  the  balance  arm.  II(^  devises  a  graphical 
method  of  making  proper  allowance  for  this  supposed  error, 
and  deduces  as  the  linal  im^an  of  all  the  experiments  of  Baily  a 
value  5.52,  the  extremes  being  5.4l>  and  5.55- 

Saigey  made  a  new  determination  of  A  (74,  vol.  VZ,  p.  377), 
from  the  ditTerence  (»".8<l  of  the  astropomical  and  geodetieal 
latitudes  of  Evaux  as  calculated  by  Puissant.  Applying  the 
method  used  by  Ilutton  for  Schehallien,  and  hiter  by  James 
and  (Jlarko  for  Arthur's  Seat,  he  found  the  ratio  of  A  to  the 
surface  density  of  France  to  be  1.7.  Assuming  the  latter  den- 
sity to  be  2.5,  the  former  becomes  4.25. 

In  1847,  liearn  tried  to  account  for  the  anomalies  in  Baily's 
results  by  assuming  a  magnetic  action.  lie  worked  out  the 
theory  (81)  of  sindi  action,  and  found  that  it  must  bo  of  a  very 
fluctuating  nature  and  may  be  either  positive  or  negative,  and 
even  greater  in  magnitude  than  the  force  of  gravitation.     That 

118 


Til  !•:    LAWS    ()|-     (;  li.\  VriATK^N 


377), 

lotical 

llamcs 
\o  the 
den- 

aily's 
It  the 
very 
»,  and 
'That 


Hiudi  a  majUMit'lic  net  ion  docs  not  icallv  rxist  is  to  \h>.  dcdncrd 
I'roni  licith's  results  willi  iron  masses  (see  paije  lit;). 

Monli<;ny  olTere«|  to  the  h'oyal  A<'adeinv  of  Ueliiiiini,  in  lS,"r.\ 
a  memoir  in  vvliieh  lu>  att rihnted  the  peeuliarit iits  in  tlii^  he- 
liaviour  of  the  torsion  pendiilnm  in  the  experiments  of  Caven- 
dish and  of  Huily  to  the  rotation  of  the  earth.  Sehaar  (S.*)),  to 
whom  the  nn'moir  was  referred  hy  llu'  Society,  proved  that  the 
rotation  (d'  the  earth  eonld  not  produce  Huch  elTects,  and  tin* 
memoir  was  not  piihlished. 

It  was  Cornii  and  liaille  who  lirst  pointiMl  out  (11'.*),  in  ISTS, 
the  main  error  in  liailv's  method.  It  lies  in  his  tal\in<'  the  Ith 
i'eadin<;  of  the  tiirnin^f-point  of  «»ne  series  of  experinn-nts  as  t  he 
Ist  of  the  next,  as  already  explainc^d.  They  shewed  that  the 
rotation  of  the  plank  Inddini;  the  massijs  coidd  not  he  per- 
formed rapidly  enoii<;h  to  ;^^'t  the  masses  into  the  new  position 
hefore  the  arm  had  l)e<^iin  its  return  journey.  They  theriffore 
rejcK'ted  the  1st  of  ea<di  series  of  4  readinii^s,  ami  cahudatetl  A 
from  the  other  '.\  in  II)  (;ases  taken  at  random  t'rom  .-tonu^  of 
liaily's  most  diveri^ent  values,  and  fouiul  r).ljir>  instead  of  r».7i;{. 
iteducinj^'  liaily's  llnal  value  in  the  same  proportion  they  ^ot 


').r)5. 


A  curious  ridation  hetween  density  and  temptu'aturo  us  pre- 
8ente(l  in  Haily's  deteriiiinations  was  pointed  out  hy  llicks 
(1(1(1),  in  18S(I.  The  nu'an  density  seems  to  faP  with  rise  of 
temperature.  The  most  |)rohahle  explamition  of  this  is  j^ivini 
by  I'oyntinj;  (LS."),  p.  5(1),  who  renuirks  that  the  exi)eriim'nts 
with  the  light  balls  happened  to  be  made  in  winter,  and  those 
with  the  heavy  balls  in  summer,  llicks  also  refers  to  several 
slight  corrections  to  l)e  made  in  Airy's  discussion  of  the  thciu-y 
— viz.,  for  the  air  displaced  by  the  attracting  nuisses,  for  the 
inertia  of  the  air  in  which  the  balls  move,  and  for  expansion 
with  change  of  temperature. 

Reich's  Skcoxd  Exi'KUIMKNT.  Ten  years  after  the  appear- 
ance of  Baily^s  memoir,  Ueich  published  (s:})  an  ac(!ouut  of 
some  further  experiments  with  his  apparatus.  In  the  begin- 
ning of  his  paper  he  pointed  out  that  Baily's  method  of  com- 
bining the  results  of  the  separate  experiments  was  better  than 
that  used  by  himself.  He  pro(!eeded  to  calculate  the  results 
of  his  first  experiments  by  IJaily's  method  and  found  for  A  the 
value  r).49±.();i0. 

tl9 


1 


w 


mi:  Mollis    ON 


ij 


r<        :! 


:j 


h«'iii;j  imjH'csst'd  with  llut  luioiiialics  in  Huily'H  «»l)s<'rviition«, 
iiiid  cspcriiilly  witli  i  lie  Viiiiatioii  of  tlif  liiiul  rcHiiits  with  tho 
ilciisity  of  \\\v  hulls,  Ucich  (JcterniiiitMl  to  repent  his  (experi- 
ments. Mis  iippiiratiis  was  set  up  this  time  in  u  see(Mi<l-story 
rnnm,  and  haily's  «h'vi<M's  were!  empioyet!  in  order  t<»  avoid  ihv 
elT('<;ts  of  tiMnperatiire  (duin^(>s.  The  only  important  change  in 
th(!  arran^cnnMit  of  the  apparatns  was  in  tin;  piacin^'  of  the  lar^o 
inaHS.  it  was  now  set  in  om;  of  fonr  depressions  tM) '  apart  in  a 
eir(;nlar  tahle  ntvolvin;^  nnder  the  halanco  ahout  a  vertical  axiu 
passinj^  tiirou^'h  the  centre  of  ontf  of  the  halls;  thns  no  corroc- 
tion  was  !iec(!ssary  for  the  attraction  of  the  tahle  and  its  sup- 
ports upon  the  hall.  The  halls  and  masses  were  those  nsed  in 
the  first  expt;riment.  Three  series  of  experiments  wcirc;  inado 
durin<;f  the  years  1817-50,  one  with  a  suspending  wire  of  thin 
copper,  one  with  thi(d\  coppitr,  and  ont;  with  a  hililar  iron  sus- 
pension.    The  final  mean  density  of  the  eartli  was  found  to  ho 

r».r)H:{-^dr.oi4'.>. 

In  order  to  make  a  test  of  I learn's  explanation  (sec  page  IIH) 
of  the  pecMiliarities  in  Haily's  results,  IJeicdi  made  sonu!  further 
experiments,  lie  ke[)t  tlu^  North  pole  of  a  stronj;  nuignct  near 
tlio  attra(!ting  lead  nniss  for  a  whole  day,  and  then  suddenly 
rotated  tho  mass  throu<j^h  ISO"  ahout  a  vertical  axis;  hut  no 
elteet  was  evident.  Hence  variations  in  the  result  arc  not  duo 
to  tho  nnignctizing  of  the  nnisses  l)y  the  earth,  or  similar 
cauvscs.  lie  then  took  off  tlu^  tin  hulls  and  suhstitutcd  success- 
ively halls  of  hismuth  and  of  iron.  The  values  of  d  wore  ro- 
spectivoly  5.5^1};^  and  5.088?  ;  the  largeness  of  the  latter  denotes 
possihly  u  diamagnetic  action  rf  tho  lead  mass;  hut  it  shews 
that  under  tho  original  circumstances  no  measurahlc  effect 
could  have  arisen  from  rmignetic  action. 

Prof.  Forhes  had  suggested  *  to  Reiidi  that  A  could  bo  found 
from  tho  period  of  the  halanco  only,  hy  noting  tho  variation  of 
tho  time  of  vibration  witU  the  position  of  the  attracting  masses. 
IJeich  nuido  some  experiments  of  this  nature  by  placing  two 
lead  masses  diametrically  opposite  to  each  other,  first  so  tha^ 
tho  lino  joining  tnem  was  perpendicular  to  tho  vertical  plane 
through  the  torsion  arm,  and  next  was  in  tho  plane.  This 
caused  no  deviation,  but  only  a  change  in  the  time  of  swing  of 


*  We  have  seen  (pajio  106)  that  this  method  was  suggested  earlier,  in 
Gehler's  Physikalische  Worterbucfi,  and  the  equations  given  by  Brandes  (42), 

120 


Til  K    LA  WS    <>K    (iU.\  VIT  ATImN 


lound 
m  of 

ISSCS. 

two 

tllilu 

|)1(inc 
JTlns 
[got 

iv,  in 

(42). 


-V-' 


tiie  Ituiiuire.  'Vhci  viiliio  of  A  foiiiKl  in  this  wity  wuh  ('i.'s','),  htit 
tlir  tippiiriit us  Wiis  not  Wfll  ilrviscd  for  \\\v  work. 

Scvcnil  ul)sliiirt8  of  Uuii'li's  |iaiK'r  nrv.  to  be  found  (84,  hO, 
ST  un.i  18.*.,  in».  :»0-',»). 

AlUV's    IJAUTON    ('OM.IKIIY     Kx  I'l:  Kl  M  KNT.     W"   llUVO  lllllMldy 

icffrrt'd  t«)  Ally's  cxpuriuiunts  in  t.ho  Dcdcoiitli  luinu  in  l8*^»>-8. 
In  IS.M,  lio  ii^iiin  undertook  to  c;uri-y  out  invcsti^itions  ( KM)) 
along  tlio  Kunic  lines,  the  introduction  of  tin;  tidei^ruph  having 
nuido  eiisy  the  eoinpurison  of  the  eloeks  iit  the  top  and  bottom 
of  the  mine,  lie  stdected  the  llarton  ('olliery,  near  South 
Shields,  for  the  experiments,  which  were  carried  out  by  six  ex- 
perienced assistants  o\'  whom  Mr.  Dunkin  was  the  chief.  The 
two  stations  were  vertically  above  each  other  a»nl  l*ir»»!  ft.  apart. 
The  apparatus  was  the  best  obtainable,  and  special  precautions 
were  taken  in  order  that  the  p(!nduliim  supports  miglit  be  rigid. 

Simultaneous  (diservations  of  the  two  petidulunis  were  kej)t 
up  night  and  day  for  a  week  ;  then  the  pendulums  were  ex- 
changed and  observations  taken  for  another  week.  Two  more 
exchanges  were  nuide,  but  the  observations  for  them  both  were 
made  in  one  week.  Each  pendulum  bad  six  swings  of  nearly 
4  hours  each  on  every  day  of  observation,  aiul  between  success- 
ive swings  the  clock  rates  were  compared  by  telegraphic  siginils 
given  every  15  seconds  by  a  journcynum  clock. 

The  corrections  and  reductions  wore  carried  out  by  Airy  in  a 
verv  elaborate  manner.  The  results  of  the  1st  and  Ikl  series 
agree  very  closely,  as  do  those  of  the  'M  and  4th,  sbowing  that 
the  pendulums  Inid  undergone  no  sensible  change.  Hy  com- 
paring the  mean  of  the  1st  and  3d  series  with  the  mean  of  the 
2d  and  4th,  the  ratio  of  the  pendulum  rates  at  the  upper  and 
lower  stations  is  obtained  independently  of  the  pendulums  em- 

Sed.      The  final  result  gave  gravity  at  the  lower  station 
Be  than  gravity  at  the  upper  by  jT^ireth  part,  with  an  un- 


V  certainty  of  ^\o^h  part  of  the  increase  ;  or  the  acceleration  of 
the  seconds-i>vHdulum  below  is  2".24  per  day,  with  an  uncer- 
tainty of  less  than  0".01. 

In  order  to  calculate  what  this  difference  should  be,  suppose 
the  earth  to  be  a  sphere  of  radius  r  and  mean  density  A,  sur- 
rounded by  a  spherical  shell  of  thickness  h  and  density  I,  then 

,         1     .     ,         ^,    ,  gravity  below      ^      2h      3//2  , 

a  simple  analysis  shews  that t-^ — r —  =  1  H r  (com- 

'■  *'  gravity  above  •        -^    \ 

131 


rA 


Iff 


MEMOIRS    ON 


!    i: 


4\ 


pure  p.  'M).  Airy  gives  n  discussion  of  tlio  effect  of  ^urfuce  ir- 
regularities ;  it  is  sliewii  that,  supposing  tiie  surfa(;e  of  tiie  eartii 
near  the  mine  bo  liave  no  irregularitie.s,  the  effect  of  tiioseat  dis- 
tant parts  of  tiie  earth  nniy  be  neglected.  lie  also  assumes  that 
there  is  no  sudden  change  of  density  just  under  the  mine,  lie 
proves  that  the  effect  of  a  plane  of  '3  miles  in  radius  and  of  the 
;,liickness  of  the  sIijII  is  'i'j  of  that  of  the  uhole  shell,  so  that 
only  the  neighbouring  country  need  be  surveyed.  Since  the 
ui)per  station  is  oidy  74  ft.  above  high  water,  it  will  be  sutllcient 
to  assume  that  any  excess  or  defect  of  matter  exists  actually  on 
the  surface.  A  careful  survey  of  the  environs  of  the  mine  was 
made,  and  allowance  made  for  each  elevation  and  depression. 
The  general  result  is  that  the  attraction  of  the  regular  shell  of 

^      .    ,     ,      T     •   •  1     1  I       1      *.     1   .1         .     gi'Ji-vity  below 
matter  is  to  be  diminished  by  about  wJ-yth  part ;  •/     , 

•*»"      1  gravity  above 

=.-1.00012O;J3-.OOO179.S4x-.      Now  from    the  pendulum  ex- 


periments Airy  found 


gravity  below 


=  1 . 00005 1 S5  ± . ()( KK »00 1 0  ; 


gravity  above 
hence  |  =  2.0'^(;(>±.OO73.     Prof.  W.  II.  Miller  found  the  aver- 

0 

age  density  of  the  rocks  in  the  mine  to  be  2.50;  hence  A  =  C.50G 
±.0l8:i. 

Airy  had  intended  that  the  temperatures  at  the  two  stations 
should  be  the  same,  but  the  temperature  of  the  lower  station 
was  7°.  13  F.  higher  than  that  of  theupper.  In  a  supplement- 
ary paper  (101)  Airy  makes  a  correction  for  this  temperature 
difference  in  two  distinct  ways,  giving  for  the  corrected  A,  G.800 
and  Ci.ij'Zd  respectively.  In  this  paper  Stokes  (102)  investigates 
tlie  effect  of  the  earth's  rotation  and  ellipticity  in  modifying 
the  results  of  the  Ilarton  experiments.  It  was  found  to  be 
small,  changing  A  from  G.50G  to  G,5G5.  *.* 

Airy  published  several  preliminary  notices  of  his  work  (88,  89 
and  122),  abstracts  of  which  appeared  in  several  journals  (1)0, 
91,  92,  98  and  111).  Valuable  re^jumes  of  the  main  paper  are 
also  to  bo  found  (105,  107,  109,  112  and  119). 

Haughton  (lOG,  110,  113  and  IIG)  gave  a  rough  but  simple 
method  of  deducing  A  from  Airy's  figures,  and  arrived  at  5.48 
as  the  value  of  A.  Knopf  (149^)  has  severely  criticized  this 
calculation.  Another  simple  formula  for  the  same  purpose 
was  given  by  an  anonymous  writer  (114).      On  the  effect  of 

122 


m 


T  U  E    L  A  W  S    O  K    (i  It  A  V  I  T  A  T  I  ( )  N 

great  clijuigcs  in  density  ])elo\v  tlic  inider  station  one  should  read 
the  paper  by  Jacob  (118  and  1*21)  already  referred  to.  Schef- 
fler  {l'^^)  pnl»'ihhed  in  IHO."),  though  it  is  dated  18r)(>,  the  pro- 
posal of  an  experiment  similar  to  Airy's,  hut  made  no  rd'erenco 
to  any  earlier  j)roj)osal8  of  the  same  kind.  Folie  (KM)  calcul- 
ated, in  1^72,  tiie  attraction  at  the  two  stations  in  a  nianncr 
diirercnt  from  Airy's,  by  considering  the  shell  as  made  up  of  "i 
parts.  Using  Airy's  data  he  arrived  at  0.4JJ0  as  the  value  of  A. 
Valuable  summaries  and  criticisms  of  Airy's  work  are  given 
by  Schcll  (i:;*)),  Zanotti-Hianco  (USJ,  pt.  '^,  pp.  UO-00),  I'oyiit- 
ing  (185,  pj).  '^4-!>)  and  Fresdorf  (180^,  p}).  13-7). 


ex- 


Jamks  and  (!lauke.  As  a  result  of  the  calculations  mad(^ 
from  the  observations  taken  for  the  Ordnance  Siirvey  of  (ireat 
Britain  and  Ireland  (104,  117,  120,  124,  125  and  120)  by  Lt. 
(Jol.  James,  it  was  found  that  the  plumb-line  was  considerably 
deflected  at  several  of  the  principal  trigonometrical  stations. 
It  was  evident  from  the  nature  of  the  ground  at  the  places 
under  consi<leration,  that  this  deflection  was  due  to  irregulari- 
ties of  the  surface.  In  order  to  study  this  action  more  care- 
fully James  decided  to  have  the  Scheliallien  experiment  re- 
peated at  Arthur's  Seat,  near  Edinburgh  (lo;}  and  125,  pp.  572- 
024).  The  observations  were  made  during  Sept.  and  Oct., 
1855,  with  Airy's  zenith -sector,  on  the  summit  of  Arthur's 
Seat  (A),  and  at  points  near  the  meridian  on  the  north  (N) 
and  south  (S)  of  that  mountain,  at  about  one-third  of  its  al- 
titude above  the  surrounding  country.  After  corrections  had 
been  applied,  the  results  were  as  follows  : 


'ing 
be 

k89 

;yo, 

are 


iple 

.48 

:his 

^ose 

of 


Station 

Astiononiical  lat.=A 

Geodetical  lat.  ~G 

A-G 

s 

55°  50'  20".  09 

55°  50'  24". 25 

2".  44 

A 

50  415  .00 

50  38  .44 

5  .25 

N 

57     9  .22 

57     2  .71 

C  .51 

It  will  be  noticed  that  even  on  the  summit  of  the  hill  there 
is  an  attraction  of  more  than  5"  toward  the  south,  which  can 
not  be  due  to  the  hill.  Similarly,  to  the  south  of  the  hill  the 
attraction  is  not  toward  the  north  jis  we  might  expect.  It  is 
evident  that  there  is  present  so!n(;  other  attracting  force,  be- 
sides that  of  Arthur's  Seat,  which  aj)pears  to  produce  a  general 
deflection  of  5"  toward  the  south, 

123 


IT 


1  '  < 
1  I 


l>l 


ii 


'A~' 


hi 


U         i 


MEMOIRS    ON 

Capt.  Clarke,  who  made  all  the  calculations,  in  order  to  find 
the  attraction  according  to  Newton's  law,  used  a  modification  of 
the  method  of  Hutton.  He  took  account  of  all  the  surface 
irregularities  within  a  radius  of  about  24000  ft.  Tiie  resulting 
value  for  the  ratio  of  the  density  of  tlie  rock  composing  the  hill 
to  that  of  the  whole  earth  was  .517JJ±.0053.  James  investi- 
gated the  density  of  the  rocks  of  Arthur's  Seat  and  found  it 
to  be  on  the  average  2.75.  This  gives  for  A  the  value  5.31G 
=h.054. 

In  order  to  see  whether  the  general  deflection  of  5"  could  be 
accounted  for  by  the  presence  of  the  hollow  of  the  River  Forth 
to  the  north  and  the  high  land  of  the  Pentland  Hills  to  the 
south,  Clarke  extended  the  calculated  attraction  to  the  borders 
of  Edinburghshire,  some  13  miles  away.  He  was  able  in  this 
way  to  account  for  a  general  deflection  of  2". 52,  and  he  thought 
that  by  carrying  the  calculations  to  Peeblesshire  the  whole  5" 
might  be  accounted  for. 

Several  abstracts  of  the  original  paper  have  been  puMis!  '^ 
(108,  115  and  123).  Poynting  (1,S5,  pp.  19-22)  has  given  a 
valuable  criticism  of  the  work. 

In  connection  with  this  investigation  might  be  mentioned 
the  various  writings  on  the  subject  of  local  attraoLions.  Any 
one  wishing  to  become  acquainted  with  this  subject  should  read 
Airy's  account  of  his  **  flotation  theory  "  (94  and  97),  Faye's 
account  of  his  '^compensation  theory"  (130,  14G|  and  147), 
Pratt's  papers  (93,  90  and  99),  Saigey  (74),  Struve  (129),  Pech- 
mann  (131),  the  treatises  of  Pratt  (133),  Clarke  (149)  and  Hel- 
mert  (148,  vol.  2).  Many  other  references  to  papers  by  these 
men  as  well  as  by  Schubert,  Peters,  Keller,  Bauernfeind  and 
others  are  to  be  found  in  the  Roy.  Soc.  Cat.  of  Scientific  papers 
and  in  Gore's  "A  Bibliography  of  Geodesy"  (174).  See  also 
note  on  page  31  and  remarks  on  page  56.  We  might  here  re- 
call the  determination  of  A  by  Saigey  from  local  attraction  (se& 
page  118).  Pechmann  (131)  in  the  same  way  found  in  the  Tyrol, 
in  1864,  two  different  values  for  A,  6.131  l=h. 1557  and  6.352 dz 
.726,  having  assumed  the  density  of  the  earth's  crust  to  be 
2.75.  We  shall  refer  later  on  to  the  determinations  of  Men- 
denhail  and  Berget. 

CoRNU  AND  Baille.  In  1873,  Cornu  and  Bailie  published 
a  short  paper  (137)  stating  that  they  had  undertaken  to  repeat 

124 


THE    LAWS    OF   GRAY  ITATION 


ipers 
also 

re  re- 
(seo 

lyi'ol, 

be 
Len- 


the  Cavendish  experiment  under  conditions  as  different  as  pos- 
sible from  those  previously  employed.  They  began  by  making 
a  thorough  study  of  the  torsion-balance  in  order  to  learn  under 
what  conditions  it  would  have  the  greatest  precision  and  sensi- 
tiveness. They  found  among  other  things  that  the  resistance  of 
the  air  was  proportional  to  the  velocity  (141, 14:^,  143  and  157). 

The  apparatus  was  set  up  in  the  cellar  of  the  ^ficole  Polytech- 
nique.  The  arm  of  the  balance  waa  a  small  aluminium  tube 
50  cm.  long,  carrying  on  each  end  a  copper  ball  109  gr.  in 
weight.  The  suspension  wire  was  of  annealed  silver  4.15  m. 
long,  and  the  time  of  vibration  of  the  system  (V  38".  The  at- 
tracting mass  was  mercury  which  could  be  aspirated  from  one 
spherical  iron  vessel  on  one  side  of  one  of  the  copper  balls  to 
another  vessel  similarly  situated  on  the  other  side  of  the  ball. 
This  method  got  rid  of  the  disturbances  arising  from  the  move- 
ment of  the  lead  masses  in  the  Cavendish  form  of  the  experi- 
ment. The  iron  vessel  was  12  cm.  in  diameter  and  the  mer- 
cury weighed  12  kg.  Another  great  improvement  was  the 
reduction  of  the  dimensions  of  the  apparatus  to  ^  of  that  used 
by  Cavendish,  Reich  and  Baily,  the  time  of  oscillation  and  the 
sensitiveness  remaining  the  same.  The  motion  of  the  arm  was 
registered  electrically. 

Two  series  of  observations  were  made ;  one  in  the  summer 
of  1872  gave  A  =  5. 56,  and  the  other  in  the  following  winter 
5.50.  The  difference  was  explained  by  a  flexure  of  the  torsion- 
rod,  and  the  former  result  was  considered  the  better. 

In  a  later  report  (142)  they  refer  to  some  changes  made  in 
their  apparatus ;  they  increased  the  force  ol  attraction  by  using 
4  iron  receivers,  2  on  each  side  of  each  copper  ball,  and  they 
reduced  the  distance  between  the  attracting  bodies  in  the  ratio 
of  V'2  to  1.  The  time  of  vibration,  408",  remained  the  same 
within  a  few  tenths  of  a  second  for  more  than  a  year.  The  new 
value  of  A  was  5.56. 

We  have  already  referred  (page  119)  to  the  fact  that  Cornu 
and  Bailie  foulul  out  the  error  in  the  Baily  experiments. 

A  final  account  of  these  experiments  has  not  yet  been  pub- 
lished. Abstracts  of  the  papers  cited  are  given  by  Poynting 
(185,  pp.  57-8)  and  by  several  journals  (138  and  139). 


Ished 
peat 


Jolly.     In  1878,  von  Jolly  of  Munich  published  an  account 
(144  and  145)  of  the  results  of  his  study  of  the  beam  balance 

125 


1 


M  H:  M  O  I  R  S    ON 


V  '• 


iij 


iiu  an  instninieiit  for  measuring  gravitational  attractions.  Tie 
disciisaod  the  sources  of  error  in  the  balance  readings  and 
methods  of  eliminating  them.  IMie  variations  due  to  tempera- 
tures etfects  are  very  ditliciilt  to  avoid,  but  by  working  in  the 
mornings  only,  and  by  covering  the  balance  case  with  another 
lined  inside  and  out  with  silver  paper,  it  was  found  to  bo  pos- 
sible to  get  ([uite  concordant  results. 

Jolly  applied  the  balanee  to  test  the  Newtonian  law  of  the 
distance.  Two  extra  scale  pans  were  suspended  by  wires  from 
the  ordiiKiry  scale  pans  of  the  balance  ami  5.^!)  tn.  below  them. 
The  wires  and  lower  scale  pans  were  enclosed  to  prevent  oscilla- 
tions from  air  currents.  Two  kilogramme  masses  of  polished 
nickel-plated  brass  were  balanced  against  each  other,  first  both 
in  the  upper  scale  pans,  and  then  one  in  the  upper  and  the 
other  in  the  lower  pan,  in  each  case  double  weighings  being  made 
after  the  manner  of  (jJauss.  The  motion  of  the  beam  was  noted 
bv  the  mirror  and  scale  method,  the  mirror  being  fixed  at  the 
middle  of  the  beam  and  perpendicular  to  its  length.  If  r  is  the 
radius  of  the  earth  at  sea-level,  and  h  a  height  above  it,  then  a 

riniss  Qi  at  sea-level  weighs  Q2  at  h,  where  Q2=Qi  (l— "    )  ap- 

Q2    1  000  000- 1.  :»()!)<) 


1  000  000 
The  differ- 


proximately.   Jolly  found  by  experiment—  = 

Qo     1  000  000-1. GG2 
whereas  the  equation  g^ves^^^       1  OOO  000 

ence,  .153  mg.,  Jolly  thought,  was  due  to  local  attractions.  He 
proposed  to  repeat  the  experiment  at  the  top  of  a  high  tower, 
and  at  the  same  time  to  find  the  mass  01  the  earth  by  noting 
the  change  in  weight  of  one  of  the  masses  in  the  balance  when 
a  large  lead  ball  was  brought  beneath  it. 

The  results  of  these  experiments  (153  and  154)  were  published 
in  1881.  The  distance  between  the  scale  pans  was  now  21.005 
m.  The  arm  of  the  balance  was  60  cm.  long,  and  the  maxi- 
mum load  5  kg.  Four  hollow  glass  spheres  of  the  same  size 
were  made  and  in  each  of  two  5  kg.  of  mercury  were  put, 
and  all  were  sealed  up.  Each  scale  pan  had  always  one  sphere 
in  it,  and  thus  Jiir  corrections  were  avoided.  An  observation 
was  made  as  follows  :  first  the  mercury-filled  spheres  were  bal- 
anced in  the  upper  pans,  and  then  one  in  the  upper  pan  was 
balanced  against  the  other  in  the  lower.     The  change  in  weight 

126 


THE    LAWS    OF    (JitA  VITATION 


TTc 


observed  was  31.G8G  mg.;  whereas  the  olian;;e  as  calciihited 
from  the  formula  sliould  have  been  33.05!)*  ing.  The  differ- 
ence is  in  tlie  same  direction  as  in  the  earlier  experiment. 

A  sphere  of  radius  .4975  m.  and  weight  5775. !;i  kg.  was  tlion 
built  up  out  of  lead  bars  under  tlie  lower  scale  pan  which 
received  the  mercury-filled  globe.  The  distance  from  the 
centre  of  this  sphere  to  that  of  the  globe  was  then  .508G  m. 
The  attraction  of  the  sphere  for  the  mc>rcury-filled  globe  when 
in  the  upper  pan  was  neglected. 

Observations  were  made  exactlv  as  before,  and  the  chanfje  in 
weight  was  3'2.^75  mg.  The  increase  in  weigiit  due  to  the 
presence  of  the  lead  is  therefore  .5(S()  mg.  Knowing  tlie  den- 
sity of  the  lead  to  be  11.18(1,  a  simple  calculation  gives  for  the 
mean  density  of  the  earth  5.(>!)"^±.(K;8. 

An  account  of  these  experiments  is  given  by  Ilelmert  (14S, 
vol.  2,  pp.  380-2),  Zunotti- Bianco  (U8|,  vol.  i,  pp.  175-8->i), 
Wallentin  (154^),  Keller  (107),  Poynting  (185,  pp.  01-4)  and 
Fresdorf  (180|,pp.  23-5). 


ap- 

l)!)<) 


Mendeniiall.  In  1880,  Prof.  T.  C.  Mendenhall  described 
(150)  a  method  of  finding  the  period  of  a  pendulum  such  that 
a  determination  required  20  or  30  minutes  only.  At  the  begin- 
ning and  end  of  this  time  the  pendulum  throws  a  light  trip- 
hammer of  wire  which  breaks  a  circuit  and  makes  a, record  on 
a  chronograph  on  which  a  break-circuit  clock  is  also  marking. 
The  advantage  of  such  an  arrangement,  in  addition  to  the 
short  time  required,  is  that  the  arc  of  vibration  may  be  small 
and  will  change  very  little.  Mendenhall  expressed  a  deter- 
mination to  find  the  variation  of  the  acceleration  due  to  gravity 
on  going  from  Tokio  to  the  top  of  Mount  Fujiyama. 

A  year  later  the  results  of  these  experiments  were  published 
(151),  having  been  made  in  Aug.,  1880.  An  invariable  pend- 
ulum was  used,  made  from  a  Kater's  pendulum  by  removing 
one  ball  and  knife-edge.  Its  period  at  Tokio  (barometer  30  in. 
and  temperatiire  23°. 5  C.)  was  .999834  sec.  On  the  top  of 
Fujiyama  the  barometer  stood  nearly  stationary  at  19.5  in. 
during  the  observations,  and  the  thermometer  at  8°. 5.  After 
approximate  corrections  were  made  for  buoyancy,  the  time, 
reduced  to  Tokio  conditions,  was  1.000330  sec.     Assuming  g  at 

*  According  to  Helmert  this  should  be  33.108  und  according  to  Zanotti- 
Bianco  33.053. 

127 


T 


™fWfp"Yi,;«  I'.jm'jm' 


I  ■  li«i  |iimwiii(|ppii*f  II  q  v^^i ,'     « 


I 

I 

i 


MEMOIRS    ON 

Tokio  to  be  Sk7084,  as  he  had  found  in  the  previous  year,  it  fol- 
lows that  at  the  summit  of  Fujiyama  it  is  0.78HC. 

No  exact  triangulation  of  the  region  had  been  made,  but 
Mendenhall  assumed  Fujiyama  to  be  a  cone  2.35  miles  high 
standing  on  a  plain  of  considerable  extent.  The  angle  of  the 
cone  was  measured  from  photograjdis  and  found  to  be  lo8°. 
Fujiyama  is  an  extinct  volcano,  said  to  have  been  made  in  a 
single  night,  and  hence  its  composition  ought  to  be  homogene- 
Oiis.  Its  average  density  was  taken  as  2.VZ,  but  no  great  re- 
liance can  be  placed  on  this  number.  Corrected  for  the  differ- 
ence in  latitude,  19',  between  Tokio  and  Fujiyama,  the  time  at 
its  base,  supposing  the  hill  taken  away,  would  be  .9!)9847  sec. 
The  density  of  the  earth,  calculated  from  these  data  after  the 
manner  of  Oarlini,  was  found  to  be  5.77. 

Fresdorf  (18C|,  pp.  11-13)  describes  fully  the  experiments 
and  points  out  an  error  in  Mendenhall's  calculations  ;  the  cor- 
rected value  for  A  is  5.CG7.  Poynting  (185,  pp.  39-40)  gives 
an  abstract  of  the  papers  referred  to. 

Stern KCK.  Major  von  Sterneck  has  made  several  investiga- 
tions of  the  variation  of  gravity  beneath  the  earth's  surface. 
The  earliest  experiments  (155)  were  made,  in  1882,  in  the 
Adalbert  shaft  of  the  silver  mine  at  Pribram  in  Bohemia.  The 
method  employ-jd  was  to  carry  an  invariable  half-second  pend- 
ulum and  a  comparison  clock  from  one  station  to  another,  and 
find  the  period  by  the  method  of  coincidences,  the  clock  being 
compared  with  a  standard  clock  by  carrying  a  pocket  chron- 
ometer from  one  to  the  other.  The  pendulum,  of  brass,  was 
a  rod  24  cm.  in  length  carrying  a  lens-shaped  bob  weighing  1 
kg.  The  knife  was  of  steel  whose  edge  was  so  cut  away  that 
it  rested  on  a  glass  plate  on  two  points  only.  The  apparatus 
was  always  enclosed  in  a  glass  case  to  prevent  air  currents. 
The  3  stations  !at  which  observations  wore  made  were  at  the 
surface,  516.0  in.  and  972.5  m.  below  the  surface  respectively. 
The  respective  periods  at  these  stations  were  .5008550,  .5008410 
and  .5008415  seconds,  and  the  resulting  values  of  A,  found 
from  Airy's  formula,  were  G.28  and  5.01,  the  density  of  the 
surface  layer  being  taken  as  2.75.  It  will  be  noticed  that  the 
values  of  ^  at  the  two  underground  stations  are  practically  the 
same,  and  the  results  are  unsatisfactory. 

A  year  later  (156)  von  Sterneck  repeated  his  experiments  at 
♦  128 


* 


TUK    LAWS    OF    UiiAVlTATlON 


)eing 


was 
ing  1 
that 
ratus 
lents. 
the 
Ivel  y. 
18410 
)uiul 
tlie 
the 
the 

bs  at 


the  same  stations  and  at  two  additional  ones.  In  order  that 
liis  observations  might  ho  independent  of  the  rates  of  the  clocks 
nsed  in  finding  tiie  periods,  Sterneck  introduced  an  important 
modification  of  tlie  metiiod  adopted  by  Airy  and  by  himself  in 
his  earlier  investigations.  lie  made  another  penduiiim  simihir 
to  the  one  described  above  ;  one  of  these  was  always  at  the 
surface  station  and  the  other  at  one  of  the  underground  sta- 
tions, and  their  rehitive  periods  were  compared  by  means  of 
electric  signals  sent  simultaneously  from  a  single  clock.  This 
clock  kept  a  circuit  closed  for  half  a  second  every  other  half 
second  and  operated  a  relay  with  a  strong  current  at  each  sta- 
tion. The  passage  of  the  *'  tail  "  of  the  pendulum  in  front  of 
a  scale  was  observed  by  means  of  a  telescope,  in  the  focal  plane 
of  which  was  a  shutter  moved  by  the  relay  current  every  half 
second,  and  at  those  instants  only  was  the  picture  of  the  tail 
of  the  pendulum  allowed  to  pass  to  the  eye  through  the  tele- 
scope. The  time  of  a  coincidence  was  when  at  one  of  these 
flashes  the  tail  appeared  exactly  at  the  middle  of  the  scale  ;  the 
time  between  two  successive  coincidences  determines  the  period 
of  the  pendulum.  The  observer  at  each  of  the  two  stations  is 
thus  finding  the  period  of  his  pendulum  ii  *;erms  of  exactly  the 
same  nnit  of  time.  When  the  observations  were  corrected,  it 
was  found  that  the  period  at  the  highest  underground  station 
was  less  than  that  at  the  next  lower  station,  and  the  determin- 
ation at  the  former  station  was  consequently  not  used.  The 
values  of  A  as  determined  from  observations  at  the  other  sta- 
tions were  5.71,  5.81  and  5.80,  with  a  mean  of  5.77.  Helmert 
(148,  vol.  2,  p.  499)  has  made  a  recalculation  and  finds  that 
these  numbers  should  be  5.54,  5.71,  5.80  and  5.71  respectively. 
Von  Sterneck  used  his  results  at  the  surface  and  at  these 
underground  stations  to  express  y  as  a  function  of  the  depth. 
Calling  the  value  of  g  at  the  surface  unity,  and  measuring  r 
from  the  centre  of  the  earth  and  calling  it  equal  lo  unity  at 
the  surface,  he  deduced  the  following  expression  for  the  value 
of  g  at  any  depth  : 

^=2.6950  r-1.8087  r'-f-.1182  r\ 
This  would  make  g  a  maximum,  l.OG,  at  r  =  .78.  The  density 
would  be  expressed  by  the  formula  (/=:15.1;)G  — 12.513  r,  giving 
15.136  for  its  value  at  the  centre  of  the  earth,  and  2.624  at  the 
surface.  These  relations  are  at  least  suggestive  if  not  con- 
vincing. 

I  .     129 


r 


i 

W  ' 

m  ' 

f 
' 

l[   ^: 

1, 

i  ' 


ME  MO  I  lis    OX 

During  tlio  year  1883  von  Sterneck  used  tlio  saino  metljoC 
and  apparatus  to  doterniino  tlio  variation  in  gravity  for  13  sta- 
tions above  tlie  earth's  surface  at  Kronstadt.  lie  found  (l')H) 
gravity  greater  at  a  higiier  point  (Schlossborg)  tlian  at  a  lower 
(Zwinger),  and  proved  that  neither  the  formula  of  Young  (see 
page  31)  nor  that  of  Faye  and  Ferrei  for  the  reduction  to  sea- 
level  gave  satisfactory  results. 

Twice  in  this  year  Sterneck  made  investigations  at  Krusna 
hora  in  Boliemia.  Here  there  was  a  mine  witii  a  horizontal 
gallery  1000  m.  long,  and  he  wished  to  find  the  effect  of  the 
overlying  sheet  of  earth  upon  the  value  of  gravity  at  various 
points  in  the  gallery.  The  same  apparatus  was  used  after  some 
im[)rovements  had  been  made.  Observations  were  taken  at  the 
mine  mouth  and  at  points  31)0  and  780  m.  from  the  mouth,  and 
G2  and  100  m.  respectively  below  the  surface  of  the  ground. 
The  results  shewed  that  gravity  in  the  plateau  increased  with 
the  depth  of  the  super-incumbent  layer  by  the  half  of  the 
amount  by  which  it  would  have  changed  in  free  space  when 
the  distance  from  the  centre  of  the  earth  was  changed  by  the 
same  amount.  Observations  were  made  at  4  stations  above 
ground  also  at  different  elevations,  and  it  was  found  that  the 
Faye-Ferrel  rule  accounted  for  the  differences  between  them 
much  better  than  did  the  Bouguer- Young  rule. 

Further  experiments  (164)  were  made,  in  1884,  at  Saghegy 
in  Hungary,  and  elsewhere,  with  results  similar  to  those  de- 
scribed above.  An  important  improvement  was  made  in  the 
method  of  observing  the  coinciilences.  They  were  now  ob- 
served by  the  reflections  of  an  electric  spark  from  two  mirrors, 
one  fixed  on  the  pendulum  stand,  and  the  other  attached  to 
the  pendulum  and  when  at  rest  parallel  to  the  first.  The  spark 
was  made  by  the  relay  circuit  every  half  second. 

In  1885,  Sterneck  made  a  series  of  observations  (165)  at  the 
mouth  and  at  4  underground  stations  in  the  Himmelfahrt- 
Fundgrube  silver  mine  at  Freiberg  in  Saxony.  He  was  led  to 
do  so  by  the  publication  of  the  results  of  some  pendulum  meas- 
urements made  there,  in  1871,  by  Dr.  C.  Bruhns,  who  had  found 
that  gravity  decreased  with  the  depth.  Using  Airy's  formula, 
von  Sterneck  found  the  following  values  for  zl  at  the  4  under- 
ground stations  in  the  order  of  their  depth  :  5.66,  6.66,  7.15 
and  7.60,  the  density  of  the  mine  strata  being  2.69.  These  re- 
sults indicate  an  abnormal  increase  of  gravity  with  depth.    Von 

130 


f  ' 


TilK    LAWS    OK    <;KAVlTATiUN 


3thO(I 

'i  sta- 
(158) 
lower 
?  (sec 

0  sea- 

rusna 
zontal 
ot  the 
arious 
1*  some 
at  the 
,h,  and 
round, 
d  with 
of  the 
e  when 
by  tlie 
above 
bat  the 

1  them 

aghegy 
ose  de- 
in  the 
3\v  ob- 
lirrors, 
hed  to 
B  spark 

at  the 

Ifahrt- 

led  to 

meas- 

found 

rmula, 

nnder- 
7.15 

lese  re- 
Von 


Sterneck  tiotioed  tluit  in  tlieso  experiments,  as  well  as  in  those 
made  at  I'rihrum,  the  increase  in  gravity  is  nearly  proportional 
to  the  increase  in  temperature.  \U\i  altliougii  i[i('i\s  (1(1(5),  as 
we  have  seen  (page  11!)),  disoovcred  a  connection  between  the 
values  of  A  and  the  tiimperaturcs  in  Ha-'y's  cxiu'riinents,  and 
Cornu  and  Bailie  (lagt*  \'i^))  got  a  larger  rcstilt  for  A  in  snmnicr 
than  in  winter,  we  have  no  reason  for  looking  ui)on  tlu*  varia- 
tions in  temperature  as  an  explanation  of  the*  afiomalics  under 
consideration.  An  interesting  criticism  of  von  Sterncck's  work 
is  given  by  Poynting  (185,  pp.  yU-3!>).  Short  a(!counts  of  it 
are  given  by  Fresdorf  (18(jJ,  pp.  17-9)  and  Giinther  (lOGJ,  vol. 
1,  p.  189). 

WiLsiNO.  In  1887,  .1.  AVilsing  (170)  made  at  Potsdam  a  de- 
termination  of  the  mean  density  of  the  earth  by  means  of  jiii 
instrument  which  is  called  the  pendulum  balance,  and  is  the 
common  beam  balance  turned  through  90°.  It  is  practically  a 
pendulum  made  of  a  rod  with  balls  at  each  end  and  a  knife- 
edge  placed  just  above  the  centre  of  gravity.  The  instrument 
used  by  Wilsing  consisted  of  a  drawn  brass  tube  1  m.  long, 
4.15  cm.  in  diameter  and  .10  cm.  thick,  strengthened  near  the 
middle  wliere  the  knife-edge  is  atlixed.  The  knife-edge  and 
the  bed  on  which  it  rested  were  of  agate,  and  (5  cm.  long.  To 
the  ends  were  screwed  the  balls  of  brass  weighing  540  gr.  each, 
and  on  the  upper  ball  was  a  pin  carrying  discs  which  were  used 
for  finding  the  moment  of  inertia  and  the  position  of  the  cen- 
tre of  gravity  of  the  pendulum.  Its  motion  was  observed  by 
the  telescope  and  scale  method,  a  mirror  being  attaclied  to  the 
side  of  the  pendulum  parallel  to  the  kjiife-edge.  The  pend- 
ulum was  mounted  on  a  massive  pier  in  the  basement  of  the 
Astrophysical  Observatory  in  Potsdam,  and  was  protected  from 
air  currents  by  a  cloth-lined  wooden  covering. 

The  attracting  masses  were  cast-iron  cylinders  each  weighing 
325  kg.  They^vere  so  arranged  on  a  continuous  string  passing 
over  pulleys  that  when  one  was  opposite  the  lower  brass  ball  on 
one  side  of  the  pendulum  the  other  was  opposite  the  upper  ball 
on  the  other  side.  Their  relative  positions  could  be  quickly 
changed  from  without  the  room,  so  that  the  former  mass  came 
opposite  the  upper  ball  and  the  latter  mass  opposite  the  lower  ; 
the  deflection  was  now  in  the  opposite  direction  from  what  it 

was  in  the  first  case. 

131 


If 


MKMolliS    ON 


Tlio  (loublo  (loflontioii  duo  to  tho  oliiin^o  in  poHJtion  of  the 
lUiisHos,  and  tlui  tiino  of  vihnitioii  art;  tlui  (|inuititic's  rcfjiiirtMl 
for  tlui  (lotcM'iniMatiou  of  A.  'V\\v.  roiidini^s  for  tlinso  (|iiiWititio.s 
woro  iimdo  by  the  mcitlnxl  of  Uuily.  which  has  hcon  iilniudy  do- 
s(M'ilK'(l.  'VUo,  linio  of  vihraLioii  was  dctiUMiiiiUMl  (irst  witii  the 
dis(!s  on  top  of  tho  ii[)|nM'  ball,  tluui  with  one  rorriovcMl  and  tinMi 
with  still  anotlxu*  rcriiovcd.  In  this  way  tlu>  moment  of  inertia 
was  obtaiiu'd.  Tho  theory  of  tho  instrumcMit  is  eomplioated, 
and  for  it  roforeiK^o  must  be  inadi!  to  tho  original  paper.  Tho 
residt  obtained  for  A  was  r».r>I)4  t.(>:i*i. 

In  ISHI),  WilsiuLj  pubiisluMJ  (W'l)  an  acoount  of  some  further 
ol)sorvations  with  tho  same  apparatus,  some  slight  changes 
baving  been  \uiu\v  in  it  in  tho  meantime.  Kxtra  pro(Miutions 
wore  taken  in  order  to  avoid  tho  elTe(!ts  of  variations  of  tom- 
poraturo.  K.\p(M'iments  svere  made  with  the  old  balls,  with  new 
load  balls,  and  with  the  pendulum  rod  alone.  Tho  mean  re- 
sult from  thes«;  was  r)..')SS±.()i;j  ;  and  tho  final  average  of  all 
his  «lotorminationH  r>.r)i!)dr.OI^. 

A  preliminary  })aper  (l(J'3)  was  read  by  Wilsing  before  tiio 
Berlin  Academy,  ami  also  an  extract  (H>1>)  of  his  first  paper. 
A  condensed  translation  of  both  papers  was  made  by  Prof.  J. 
II.  (Joro  (ITl)  for  the  Smithsonian  Report  for  1888,  and  a  short 
account  of  the  work  is  given  by  Poynting  (185,  pp.  G5-9)  and 
by  Fresdorf  (KSOJ,  p.  28). 


I 


r 


PoYNTrxo.  Prof.  J.  II.  Poynting  published  in  1878  the 
results  (140)  of  a  study  of  tho  beam  balance.  He  found  tliat 
tho  sources  of  error  were  temperature  changes  producing  con- 
vection currents  and  unequal  expansion  of  the  arms,  and  tlie 
necessity  of  frequently  raising  the  knife-edges  from  the  planes. 
Ho  tried  to  overcome  tlie  former  difficulty  by  taking  the  same 
precautions  as  those  employed  by  users  of  the  torsion  balance  ; 
and  he  did  away  altogether  with  the  raising  of  the  beam  be- 
tween weighings,  and  when  the  weights  had  to  be  exchanged 
held  the  pan  tixed  in  a  clamp. 

The  paper  gives  a  description  of  his  balance  and  illustrates 
how  it  can  be  used,  (1)  to  compare  two  weights,  and  (2)  to  find 
the  mean  density  of  the  earth.  The  motion  of  the  beam  was 
observed  by  means  of  a  telescope  and  scale,  the  mirror  being 
fixed  at  the  centre  of  the  beam.  The  deflection  of  the  ray 
could  be  multiplied  by  repeated  reflections  between  this  mirror 

132 


TIIK    r.AWS    (»F    <;KAV  ITATIoN 


and  unotlicr  whicli  wiw  flxod  jiiid  nearly  paralli'l  to  tlic  fortnor. 
'I'lio  ceiitro  of  oscillation  was  (U'tciiniiitd  after  tlio  tnetliod  of 
hiiily  with  tlio  torsion  l)alan<*<'.  As  a  result  of  II  observations 
I'rof.  I'oyntini;  found  tln^  mean  density  o1  tint  earth  to  ho 
A.OOdi.lo.  ll(}  felt  jiistilied,  therefore,  in  proceeding,'  to  havi^  a 
Miort5  Huitahlo  halancis  constructed  in  order  to  niake  a  more 
careful  determination  of  this  <|uantity. 

The  inv(!stigation  (Mnitinued  through  many  years,  and  the 
results  (I8<)autl  IH."»,  j)[».  TI-IT)*;)  were  not  puhlished  until  IH'.M. 
Many  unforeseen  diniculti(\s  arose  during,'  the  pro^^ress  of  the 
work,  hut  by  patience  and  skill  I*oyntin<^  was  al)le  to  overcome 
these  difli(!ulties  and  to  be<jfin  to  take  observations  in  18!M). 
Tiie  balance  was  of  the  larj^'e  bullion  type,  Vrl'.i  cm.  long,  and 
made  with  extra  rigidity  by  Oertling.  It  was  set  up  in  a  base- 
ment room  at  Mason  (/(dlege,  liirmingham.  The  prin(Mple 
upon  whieh  the  experiment  is  based  is  as  follows  :  two  balls 
of  about  the  satno  mass  are  suspended  from  the  two  arms  of 
the  balance.  Beneath  the  balance  is  a  turn-table  carrying  a 
heavy  spherical  mass  vertically  under  one  of  the  balls.  The 
position  of  the  beam  is  observed  and  the  turn-table  moved  un- 
til he  mass  is  nnder  the  other  ball  and  the  position  of  the  beam 
again  observed.  The  deflection  measures  twice  the  attraction 
of  the  mass  for  the  ball.  The  attra(!tion  of  the  mass  for  the 
beam  and  wires,  etc.,  is  then  eliminated  by  repeating  these  ob- 
servations with  the  balls  suspended  at  a  different  distance  be- 
low the  arm,  for  then  the  attraction  of  the  mass  on  the  balance 
remains  the  same,  and  we  find  the  change  in  the  attraction  of 
the  mass  for  the  ball  with  change  oi"  distance.  The  calculation 
was  complicated  by  the  presence  on  the  turn-table  of  another 
mass  as  a  counterpoise  to  the  former  one  ;  it  was  snuiller  than 
that  one  and  at  a  correspondingly  greater  distance  from  the 
centre.  It  was  used  because  certain  anomalies  could  be  ac- 
counted for  only  on  the  supposition  that  the  floor  tilted  when 
the  turn-table  was  rotated  with  the  large  mass  only  upon  it. 

Instead  of  the  ordinary  mirror  fixed  on  the  beam,  Poynting 
used  the  double-suspension  mirror  (see  Darwin  B.  A.  Rep., 
1881).  The  riders  were  manipulated  by  mechanism  from  with- 
out,  and  the  observer  was  stationed  in  the  room  above,  whence 
he  could  make  all  changes  and  observations  without  opening 
the  balance  room.  The  attracted  and  attracting  masses  were 
made  of  an  alloy  of  lead  and  antimony.     The  balls  were  gilded 

183 


mr^-t^    II  <    ^  I 


If    i. 


MKMolltS    ON 

ftnd  woi^liod  ovor  '^1  (HK>  ^r.  cucli.  'I'lin  lurgo  masH  wei^.  „vt 
l5(MMK)^ri'.,  ami  til,.  (!omit('i|)(>is»' jilxmt  Imlf  iis  niiicli.  A  MrHt 
Hot  of  ()l)Ht'rvat  ions  pivo  A  =  .*>.")"i.  'I'lio  iittnuititi^  ImmUoh  wore 
tlu'ii  all  iiiviM'tcd  ill  onicr  to  (>liiiiiimt(t  tlio  t'tTtu^ts  of  wunt  of 
Hyntiiietry  in  tlu^  ponitioii  of  tlio  tiini-tablts  and  of  lioino^e- 
nt'ity  ill  tlio  masses.  A  now  sc^t  of  ol)si'rvatioiiH  ^av(^  ^^5.40. 
The  ililTercnco  bc'tweeii  tlie  results  of  tlie  two  sot8  must  havo 
been  caustMl  by  a  cavity  or  irre<;ular  distrihiitioii  of  density  in 
tlie  iar^M^  mass,  and  l»y  other  experiments  l*rof.  I'oynting  found 
that  its  (U'litro  of  gravity  was  not  .»i>  its  centre  of  ligiiro,  but 
was  nearly  at  tli(»  place  at  which  his  gravitational  experiments 
would  have  suggested  it  to  be.  The  mean  result  for  A  is  taken 
to  lu^  r».4pJU,  and  for  \\w  gravitation  constant,  (i,  0.('>!)H4x  10-^ 

I'oynting  remarks  that  the  effects  of  (!onvection  (lurrents  are 
greater  in  the  beam  balance  than  in  the  torsion  balance,  since 
the  motion  of  the  former  is  in  a  vertical  plane.  lie  thinks 
that  a  balaiKH!  of  greatly  reduced  dimensions  would  have  been 
preferable.  The  admirable  way  in  wliich  Prof.  Poynting  has 
utilized  tlie  common  balance  for  absolute  measurements  of 
force  caused  the  University  of  Cambridge  to  award  him  the 
Adams  l*rize  in  ISW.i. 

For  a  short  account  of  this  work  see  Wallentin  '^54J). 


It. : !-'  ■ 


Reikjkt.  In  1750,  Hougujr  read  before  the  ^icademy  of 
Sciences  the  results  (0)  of  some  experiments  made  by  him  to 
determine  whether  the  plumb-line  was  affected  by  the  tidal 
motion  of  the  ocean.  He  was  not  able  to  detect  any  such 
effoct.  Towards  the  middle  of  last  century  Boscovitch  pro- 
posed (140,  vol.  1,  pp.  314  and  327)  to  phme  a  long  ])endulum 
in  a  very  high  tower  by  the  edge  of  the  sea,  where  the  height 
of  the  tide  is  very  great,  and  to  observe  the  deviation  due  to 
the  rise  of  the  water,  and  thence  to  calculate  the  mean  density 
of  the  earth.  Von  Zach  suggested  (40,  vol.  1,  p.  17)  a  modification 
of  the  experiment.  Boscovitch  also  proposed  the  use  of  a  reser- 
voir after  the  manner  about  to  be  described,  used  by  Berget. 
In  1804,  Robison,  in  his  ''Mechanical  Philosophy''  vol.  1,  page 
339,  points  out  that  a  very  sensible  effect  on  the  value  of  grav- 
ity might  be  observed  at  Annapolis,  Nova  Scotia,  due  to  the 
very  high  tides  there.  The  theory  of  this  local  influence  is 
given  in  Thomson  and  Tait's  "Natural  Philosophy  "  pt.  II., 
page  389.     Struve  (129)  proposed  to  find  A  from  observations 

134 


I 


TllK    LAWS    OF    GKAVITATIUN 

«»ri  plumb-lines  plucod  on  ouch  aide  of  the  liristol  Clunmel,  uful 
Keller  (UIH)  culculiited  tlie  <lefleetion  of  the  pliirnh-liiie  due 
to  the  dniiiiiii^  of  Luke  Fuciiio.  In  IH'.KJ,  M.  lierj^et  iitilizecl 
thin  principle  in  order  to  find  (IHI)  the  denHity  of  the  oiirth. 

lie  hud  the  nne  of  u  hike  of  '.\'i  he(;tures  in  un>u  in  the  Com- 
niiino  of  Iluhuy-lu-ncuve  in  ikd^ian  Luxeinhour^.  The  levtd  of 
the  hike  eoiild  he  lowered  1  in.  in  u  few  hourw,  iind  hh  ({uickly 
regained.  lie  could  thus  introduce  under  his  iiiHtrunient  a 
I)ra(^ticiilly  infinite  phino  of  nuitter  whose  iittriiotion  couhl  he 
ctilcu lilted  and  observed.  The  apparatus  used  to  rneaHuro  the 
attraction  was  the  hydrogen  graviineter  such  as  Boussingault 
and  Mascart  (('onip.  Uend.,  vol.  !)r),  j)p.  I'^d-S)  used  to  find  the 
diurnal  variation  of  gravity.  The  variation  of  the  column  of 
mercury  was  observed  by  the  interference  fringes  in  vacuo  be- 
tween the  surface  of  the  mercury  and  tlie  bottom  of  the  tube, 
\^'hich  was  worked  o])tically  plane.  A  first  series  of  observa- 
tions was  made  when  the  lake  was  lowered  50  cm.,  and  another 
when  it  was  lowered  1  n).  A  change  of  1  m.  caused  a  displace- 
ment of  the  mercury  column  of  l.ijOx  10-«  cm.  The  value  of 
the  gravitation  constant  found  was  O.80xl()~^  of  A,  5.41  and 
of  the  mass  of  the  e;i'  th  5.85  x  10"  gr. 

M.  (louy  remarks  (182)  that  such  a  result  would  iniply  that 
the  temperature  remained  constant  during  hours  to  y  ttfJ  (TTnr 
of  a  degree,  which  is  impossible.  Pavilion,  with  the  greatest 
care,  was  able  to  reach  xirVinr  of  a  degree  only.  So  that  the 
result  given  by  lierget  can  not  be  so  accurate  as  he  supposed. 

For  )i  short  account  of  the  experiment  see  Fresdorf  (186^, 
pp.  29-30). 

Boys.  Prof.  C.  V.  Boys  read  before  the  Royal  Society,  in  1880, 
an  important  paper  (175  and  176)  on  the  best  proportions  and 
design  for  the  torsion  balance  as  an  instrument  for  finding  the 
gravitation  constant.  He  shewed  that  the  sensibility  of  the 
apparatus,  if  the  period  of  oscillation  is  always  the  same,  is  in- 
dependent ofs  the  linear  dimensions  of  the  apparatus  ;  and  re- 
marked that  the  statements  of  Cornu  on  this  point  (page  125) 
are  not  correct.  There  are  great  advantages  to  be  gained  by 
reducing  the  dimensions  of  the  apparatus  of  Cavendish  50  or 
100  times ;  the  main  one  is  that  the  possibility  of  variation  of 
temperature  in  the  apparatus  is  enormously  minimized.  Then, 
too,  the  case  can  be  made  cylindrical  and  corrections  for  its  at- 

185 


iMKMOiU.S    (IN 


;i.i 


h-t- 


traction  avoided.  Until  quartz  fibres  existed  it  would  have 
been  impossible  to  have  made  this  reduction  in  the  dinuMisions 
of  the  apparatus  and  retained  the  period  of  5  to  10  minutes. 
The  introduction  of  tiiis  invaluable  new  moans  of  suspension 
is  also  due  to  I^rofessor  Boys.  Another  improvement  in  the 
form  of  the  apparatus  devised  by  him  is  the  suspending  of  the 
small  bails  at  diiferent  distances  below  the  arm  (the  masses 
must  bo  at  corresponding  levels),  so  that  each  mass  acts  prac- 
tically on  one  ball  only. 

Boys  showed  to  tho  Society  a  balance  of  this  design  ;  it  l»ad 
an  arm  of  only  13  mm.  in  length,  was  18.7  times  as  sensitive  as 
that  of  Cavendish  and  behaved  very  satisfactorily.  He  pro- 
posed to  prepare  a  balance  of  this  kind  especially  suitable  for 
absolute  determinations  and  capable  of  determining  the  gravi- 
tation constant  to  1  part  in  10  000. 

An  account  of  his  completed  work  (187  and  181))  was  read  in 
1804.  For  the  details  of  this  beautiful  experiment  and  the  in- 
genious way  in  which  the  apparatus  was  designed  the  original 
paper  must  be  consulted.  The  general  design  was  that  of  his 
earlier  apparatus,  but  very  great  attention  was  given  to  the 
minutest  details,  and  especially  to  the  arrangements  for  meas- 
uring the  dimensions.  Some  idea  of  the  accuracy  aimed  at 
may  be  got  from  considering  that  in  order  to  obtain  a  result 
correct  to  1  in  10  000  it  was  necessary  to  measure  the  large 
masses  to  1  in  10  000,  the  times  to  1  in  20  000,  some  lengths  to 
1  in  20  000  and  angles  to  1  in  10  000.  The  dimensions  finally 
used  were,  diameter  of  masses  2.25  and  4.25  in.;  distance  be- 
tween masses  in  plan  4  and  6  in.  ;  distance  between  balls  in 
plan  1  in. ;  diameter  of  balls  .2  smd  .25  in. ;  diflference  of  level 
between  upper  and  lower  balls  «  in.  The  masses  were  of  lead 
formed  under  great  pressure,  and  the  balls  of  gold. 

The  moment  of  inertia  of  the  beam  was  determined  by  find- 
ing the  period  when  the  balls  were  suspended  from  it,  and 
when  they  were  taken  away  and  a  cylindrical  body  of  silver, 
equal  in  weight  to  the  balls  with  their  attachments,  suspended 
from  the  middle  of  the  beam.  The  apparatus  was  enclosed  by 
a  series  of  metallic  screens  to  prevent  temperature  changes, 
and  outside  of  all  was  a  double -walled  wooden  box  with  the 
space  between  the  walls  filled  with  cotton-wool.  The  final  result 
for  the  gravitation  constant  was  6.6576  x  10~^  and  for  ^,  5.5270. 
The  last  figure  in  each  ca&e  has  no  significance,  but  Boys  con- 

136 


I 


TllK    LAWS    OF    GRAVITATION 


Ind- 
land 
ver, 
Ided 

by 

jes, 
the 
uilt 
570. 
ton- 


sidored  that  the  next  to  tlie  hist  couhl  not  he  more  tlian  2  in 
error  at  the  outside,  lie  is  still  ronviiiced  that  I  part  in 
10  000  can  be  reached,  but  would  increase  the  length  of  tin; 
beam  to  5  cm.,  since  the  disturbing  moments  due  to  convection 
are  proportioual  to  the  r)th  power  of  the  linear  dimensions,  not 
to  the  7tli  as  he  liad  originally  supposed.  An  excellent  resume 
of  the  experiment  is  to  be  found  in  the  lecture  delivered  by 
Boys  before  the  Royal  Institution  (188). 

EoETVOES.  A  series  of  investigations  upon  g»"ivitation  is  now 
under  way  by  Prof.  R.  von  Eotvcis  of  Budapest.  He  has  pub- 
lished a  preliminary  account  (H):i)  only  of  his  experiments,  but 
they  promise  to  be  very  elaborate  :i;id  exhaustive.  His  paper 
begins  with  a  mathematical  discussion  of  the  space -variation 
of  gravity  as  deduced  from  the  potential  function.  He  investi- 
gates the  equipotential  surface  and  the  measurements  necessary 
to  determine  tlie  principal  radii  of  curvature,  the  variation  of 
gravity  along  the  surface,  and  the  variation  perpendicular  to  the 
surface.  The  latter  has  already  been  measured  with  the  pend- 
ulum, and  by  Jolly  (144  and  145),  Keller  (15'^),  Thiesen  (171)) 
and  others  with  the  common  balance.  For  the  measurement 
of  the  other  quantities  von  Eutvos  uses  the  torsion  balance. 
This  he  makes  in  two  forms :  the  first  is  of  the  same  general 
type  as  that  of  Baily  and  is  called  the  K'7'ummfut(/sr((ri<miefcr, 
since  it  is  used  to  measure  the  difference  of  the  reciprocals  of 
the  principal  radii  of  curvature ;  the  second  is  like  that  of 
Boys  in  that  one  ball  is  on  one  end  of  the  rod  and  the  other 
suspended  100  cm.  below  the  other  end  by  means  of  a  wire, 
and  is  called  the  Horizontalvarionieter.  The  peculiarity  of 
these  instruments  is  in  the  method  devised  for  getting  rid  of 
convection  currents  ;  von  Eotvos  makes  the  case  with  double 
walls  of  thin  metal  with  an  air-space  of  from  5  to  10  mm.  So 
steady  is  the  motion  that  the  balance  can  be  used  i?i  any  room 
in  the  laboratory,  and  even  in  the  free  air  at  night.  The  period 
is  usually  froiA  10  to  20  minutes,  and  the  suspension  wire  is  of 
platinum  of  100  to  150  cm.  length.  The  rod  swings  in  a  flat 
cylindrical  box  40  cm.  in  diameter  and  2  cm.  deep. 

Some  investigations  have  been  made  of  the  variation  of 
gravity  in  the  neighbourhood  of  the  hill  S^ghberg,  where  von 
Sterneck  found  great  peculiarities.  Some  preliminary  deter- 
minations have  been  made  of  the  constant  of  gravitation  also, 

137 


f 


I  i- 


['    i 


MEMOIRS    ON 

with  a  result  6.05xl0~^  Von  Eotvos  speaks  of  tlie  method 
cr,ploye(]  as  jin  entirely  new  one,  but  it  is  only  a  variation  of 
that  already  einj)loyed  by  Reich,  and  later  by  Dr.  Braun,  the 
oscillation  method.  The  instrumeTit  (the  Kriimmungsvariom- 
eter)  is  set  up  between  two  piUars  of  lead,  and  the  time  of  vib- 
ration observed  both  when  the  torsion  rod  is  in  the  line  joining 
the  pillars  and  when  it  is  perpendicular  to  this  line.  The 
paper  is  characterized  by  an  almost  total  disregard  of  the  work 
already  done  in  the  field  of  gravitation. 

Eotvos  gives  a  description  of  two  new  instruments  for  use 
in  the  study  of  gravitatioil.  One  he  calls  the  Graviintioncom- 
penmtor ;  in  design  it  is  similar  to  the  others,  but  tde  arm 
swings  in  a  narrow  tube.  The  tube  is  surrounded  at  each  end 
by  the  compensating  masses  having  the  balls  at  their  centres ; 
tliese  masses  are  of  the  shape  of  a  disc  with  two  almost  com- 
plete quadrants  taken  away,  just  enough  being  left  to  hold  the 
remaining  two  quadrants  together.  By  orienting  these  masses 
any  amount  of  compensating  attraction  required  can  be  pro- 
duced. The  other  instrument  is  called  the  Gravitationmulti- 
plicator ;  underneath  the  torsion  balance  is  a  turn-table  with 
the  attracting  mass ;  when  the  ball  has  reached  its  maximum 
elongation  in  the  direction  of  the  mass,  the  latter  is  suddenly 
moved  to  the  opposite  side,  and  so  on.  From  the  difference 
of  two  successive  elongations,  and  a  knowledge  of  the  damp- 
ing, the  amount  of  the  attraction  can  be  determined.  This 
is  rather  similar  to  a  piece  of  apparatus  proposed  by  Joly  (177), 
in  1890. 


Braun.  One  of  the  latest  and  most  elaborate  determinations 
of  the  mean  density  of  the  earth  is  that  made  by  Dr.  Carl 
Braun,  S.J.,  at  Mariaschein  ^n  Bohemia  (193).  In  its  gen- 
eral form  the  apparatus  is  like  that  employed  by  Reich  in  his 
later  experiments.  Like  Reich,  too,  he  uses  two  distinct 
methods  of  finding  his  results,  the  deflection  and  the  oscilla- 
tion methods.  The  experiments  of  Dr.  Braun  differ  however 
in  several  very  important  respects  from  those  of  Reich ;  the 
dimensions  of  the  apparatus  are  much  reduced,  the  masses  are 
suspended  from  wires  and  the  deflection  is  determined  differ- 
ently;  but  the  respect  in  which  it  differs  from  all  previous  de- 
terminations is  in  the  fact  that  the  torsion  rod  swings  in  a 
partial  vacuum  of  about  4  mm.  of  mercury.     This  was  sug- 

138 


THE    LAWS    OF    GRAVITATION 


gested  by  Faye  (130)  anrl  by  Boys,  but  no  investigation  of  the 
kind  had  ever  been  made. 

The  experiments  were  begun  in  1887.  In  a  corner  of  a  liv- 
ing-room a  heavy  stone  slab  was  set  into  the  stone  wall ;  on 
this  was  a  glass  plate  from  wliieh  arose  a  brass  tripod  to  carry 
the  suspension  wire,  1  m.  long,  and  the  torsion-rod  ;  and  on 
the  plate  fitted  airtight  a  conical  glass  cover  within  which  a 
vacuum  could  be  made.  The  apparatus  was  so  tight  that  the 
pressure  inside  did  not  change  in  4  years.  Suspended  from  a 
movable  ring  encircling  the  glass  cover  were  the  two  masses, 
about  42  cm.  apart ;  the  masses  used  were  two  sets  of  spheres, 
one  of  brass  weighing  5  kg.  each,  and  the  other  of  iron,  11'^  mm. 
in  diameter,  filled  with  mercury,  and  weighing  about  0.15  kg. 
each.  The  torsion  rod  was  a  triangle  of  coj)per  wires  and  the 
balls  were  suspended  from  its  ends  and  lay  in  the  same  hori- 
zontal plane  24.6  cm.  apart.  Each  ball  was  of  gilded  brass  and 
weighed  about  54  gr.  In  order  to  provide  against  temperature 
changes,  the  whole  apparatus  was  surrounded  by  metal  screens, 
cloth  hangings  and  wooden  enclosures. 

The  deflection  method  of  observation  is  practically  that  of 
Cavendish,  but  the  position  of  the  centre  was  found  rather 
differently.  Dr.  Braun  observed  the  time  of  the  passage  in 
each  direction  across  the  wire  of  the  telescope  of  several  scale 
divisions  near  the  centre,  and  took  as  the  centre  that  point 
with  reference  to  which  the  time  of  oscillation  was  the  same  in 
both  directions.  He  found  for  the  final  corrected  mean  result 
A  =  5-.  52962. 

In  the  oscillation  method  the  period  was  determined  when 
the  masses  were  in  the  line  joining  the  balls,  and  also  when  they 
were  in  a  line  at  right  angles  to  that  direction.  The  final  cor- 
rected mean  result  gave  A  =  5. 52020.  The  mean  of  all  is 
5.52945±.0017.  The  extremes  were  5.5004  and  5.5511.  The 
mean  of  all  the  results  found  in  1892  was  5.52770,  and  in  1804 
was  5.53048. 

The  final  result  for  the  gravitation  constant  was  6.655213 
X 10-8. 

In  an  appendix  is  given  a  further  discussion  of  the  correc- 
tions to  be  made  on  account  of  damping.  Dr.  Braun  found  his 
former  estimate  to  be  in  error,  and  after  lamination  gave  as 
the  most  probable  final  results,  A  =  5.52700±:.0014,  and  G 
=(6. 65816  ±.00168)  x  lO-^. 

139 


H 

I  ! 

ii! 


il 


i 


I*; 


>  I 

1; 


It. ' 


E«  » 


r  1 


M  K  M  ()  I  II S    ()  N 
A  concise  Jiccoinit  of  the  work  la  given  in  Nature  (107). 

KoKNio,  Ru'HAiiz  AND  KuiUAU -  Mknzki..  Ill  1884,  Pro- 
fessors A.  Konig  sind  V.  Uidiiirz  proposed  (150  und  UiO)  to 
doterniine  tlie  gnivitiition  constjint  by  a  nietliod  wliicli  is  ii 
niodificution  of  tlisit  used  by  Jolly  (page  Vlij).  In  the  hitter 
experiment  the  h)\ver  set  of  scale  pans  was  'i\  in.  beneatli  the 
upper  and  ditTerences  of  temperature  were  unavoidable.  Tiie 
improvement  proposed  was  to  have  the  sets  of  scale  pans  much 
closer  together,  to  measure  the  cliange  of  weight  with  height 
after  the  manner  of  Jolly  and  then  to  insert  between  the  upper 
and  lower  pans  a  huge  block  of  lead  with  holes  in  it  for  the 
passage  of  the  wires  supporting  the  lower  pans.  A  weighing 
was  made  with  two  nearly  equal  masses,  one  in  the  right  upper, 
the  other  in  the  left  lower  pan  ;  then  the  former  in  the  right 
lower  is  balanced  against  tlie  latter  in  the  left  upper  pan.  From 
these  weighings,  taking  account  of  the  result  of  similar  ob- 
servations without  the  block,  the  value  of  4  times  the  attrac- 
tion of  the  block  is  determined,  and  from  a  comparison  of  this 
result  with  the  calculated  attraction,  the  gravitation  constant 
can  be  determined.  Professors  Konig  and  Richarz  seem  to 
have  hit  upon  the  same  idea  independently  of  each  other.  In 
1881,  Keller  proposed  (1H7)  a  somewhat  similar  modification  of 
the  Jolly  experiment.  Professor  A.  M.  Mayer  suggested  (101) 
the  use  of  mercury  instead  of  lead  for  the  attracting  mass,  but 
Konig  and  Richarz  replied  (1<)'^)  that  Mayer  had  misunder- 
stood the  forin  of  the  experijuent,  and  gave  a  lucid  and  simple 
explanation  of  their  method. 

In  1803,  appeared  a  report  (183  and  184)  on  the  observations 
made  to  find  the  decrease  of  gravity  with  increase  of  height. 
A  description  is  given  of  the  balance  and  of  the  improvements 
introduced  into  it  in  order  to  overcome  the  liability  to  varia- 
tion in  its  readings.  The  masses  weighed  against  each  other 
were  1  kg.  each,  and  the  balance  had  a  sensitiveness  of  1  part 
in  1  000  000.  All  exchange  of  weights  was  made  au^^omatically 
without  opetiing  the  covers.  The  apparatus  was  carefully  sur- 
rounded with  metal  screens  to  ward  off  temperature  changes. 
It  was  set  up  in  a  bastion  of  the  citadel  at  Spandau,  and  in 
consequence  of  the  departure  of  Konig  to  accept  a  professor- 
ship at  Berlin,  Dr.  Krigar-Menzel  assisted  in  the  carrying  on 
of  the  research.     The  pans  were  '^.20  m.  apart  vertically.    The 

140 


3"'! 


THK    LAWS    OF    GRAVITATION 


m. 


whereas  the 


01) 

n\t 

er- 

iple 


iilly 


in 

for- 

011 

^he 


change  in  ffnivitv  observed  was  .0000005:^3  , 

cjilculjited  value  was  .O0OOOG07.  The  difToronce  is  ascribed  to 
tlie  local  attraction  of  walls,  etc. 

The  paper  [V.^S)  embodying  the  final  results  was  presented 
to  the  Berlin  Academy  in  Dec,  1897.  A  most  exhanstive  ex- 
amination had  been  made  of  the  jiossible  sources  of  error,  and 
the  devices  for  overcoming  these  difficnlties  were  most  ingen- 
ious and  elaborate.  In  the  cases  where  the  sources  of  error 
could  not  be  eliminated,  as  in  the  variations  of  temperature  with 
time  and  place,  the  otfect  is  carefully  considered  and  allowed 
for.  Observations  were  made  continuously  from  Sept.,  18!M),  to 
Feb.,  181M),  and  from  the  elegance  of  the  method  and  the  time 
and  care  devoted  to  the  working  out  of  the  result,  this  deter- 
mination of  the  gravitation  constant  and  the  mean  density  of 
the  earth  must  be  taken  as  one  of  the  very  best. 

The  block  of  lead  weighed  100  000  kg.,  was  '^00  cm.  high  and 
JilO  cm.  square  and  was  built  up  out  of  bars  of  lead  10  x  10x30 
cm.  on  the  top  of  a  massive  pier.  The  amount  of  the  settling 
of  the  pier  was  measured  and  found  to  be  not  important,  and 
the  shape  of  the  block  was  not  distorted  by  its  own  pressure. 
The  final  value  for  U  was  (0. 085 ±.011)  10-s,  and  for  A.  5.505 
±.009. 

Professors  Richarz  and  Krigar-Menzel  published  (191  and 
199),  in  1890,  f  condensed  account  of  their  work.  Other  ab- 
stracts are  also  to  be  found  (185,  pp.  04-5,  180^,  pp.  20-7,  and 
194). 

Minor  Notices.  In  1889,  Dr.  W.  Laska  of  Prague  proposed 
(173)  a  method  of  finding  the  density  of  the  earth.  At  the  toj) 
of  a  rod  projecting  above  a  pendulum  is  a  lens  which  is  so  close 
to  a  fixed  plate  of  glass  that  Newton's  rings  are  visible.  A 
hollow  ball  near  the  bob  of  the  pendulum  is  then  filled  with 
mercury  and  attracts  the  bob,  bringing  the  lens  nearer  the 
plate  ;  an  observation  of  the  movement  of  the  Newton's  rings 
will  measure  the  deflection  of  the  bob.  No  further  report  has 
been  published.  An  account  of  his  method  is  given  by  Gttnther 
(19G|,  vol.  1,  p.  197). 

About  the  same  time  Professor  Joly  of  Dublin  suggested 
(177)  a  resonance  method  for  the  same  purpose.  A  pendulum  in 
a  vacuous  vessel  has  the  same  ])eriod  as  two  massive  ones  kept 

141 


M  KM  0  1  IIS    ON 


going  outsiile  tlio  vessel.  The  amplitude  of  tlie  motion  of  the 
inner  pendulum  duo  to  a  given  number  of  swings  of  the  outer 
ones  would  give  a  measure  of  the  constant  of  gravitation. 

In  1895,  Professor  A.  S.  Mackenzie  of  Bryn  Mawr  College 
publislied  an  account  (100)  of  some  experiments  with  the  Boys' 
form  of  torsion  balance  to  determine  whether  the  gravitational 
properties  of  crystals  vary  with  direction.  No  such  variation 
was  found  in  the  case  of  calc-spar,  the  crystal  under  investiga- 
tion. He  shewed  further  that  the  inverse  square  law  holds 
good  in  the  neighbourhood  of  a  crystal  to  one-fifth  per  cent. 

Two  years  later  ai)peared  an  account  (190)  of  an  investigation 
by  Professors  Austin  and  Thwing  of  the  University  of  Wiscon- 
sin to  determine  whether  gravitational  attraction  is  indepen- 
dent of  the  intervening  medium,  that  is,  whether  there  is  a 
gravitational  permeability.  No  effect  was  found  due  to  the 
medium  within  the  limits  of  error  of  the  method. 

At  a  meeting  of  the  '*  Deutcher  Natnrforscher  und  Aerzte" 
in  Brunswick,  in  1807,  Professor  Drude  read  a  paper  (195)  on 
action  at  a  distance,  wiiicli  contains  a  very  valuable  account  of 
the  theory  of  gravitation,  and  should  be  consulted  by  any  one 
wishing  to  find  a  brief  resume  of  that  subject,  and  especially 
for  a  discussion  of  the  velocity  of  propagation  of  gravitation. 

The  latest  work  on  the  laws  of  gravitation  is  that  of  Profess- 
ors Poynting  and  Gray  (200)  on  the  search  for  a  directive 
action  of  one  quartz  crystal  on  another.  A  small  crystal  was 
suspended  and  its  time  of  rotative  vibration  noted  ;  a  large 
crystal  in  the  same  horizontal  plane  was  then  rotated  about 
a  vertical  axis  through  its  centre  with  a  period  either  equal  to, 
or  twice,  that  of  the  smaller  crystal.  If  there  were  any  directive 
action  the  small  crystal  should  be  set  in  vibration  by  forced 
oscillations ;  no  such  effect  was  found. 


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II; 


THE    LAWS    UF    GKAVITATIUN 


BIBLIOORAPITY 


^■2 

SS 
-H-H 
S| 

u:  o 
II  II 
->? 

a 

io 

s 

c 
u 

a 
o 

a 


•o 


S 

p 


3 

to 

3 

es 


c 
S 


a 

3 

2 

CQ 


1 

1600 

W.  Gilbert. 

2 
3 

1665 
1687 

P.  Bftcon. 
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4 
5 
6 

7 
8 

1705 
1727 
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1749 
1751 

R.  Hooke. 
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9      1754    P.  Bouguer. 


10      1756    T.  Birch. 
11*    1769    J.  Coultaud. 


12* 

13 

14* 


1769 


15*    1771    Mercier. 


16*    1772    G.  L.  Lesage. 


18*    1772 


19      1773 


20      1773 


I 


21  1773 

22  1773 

23  1774 

24  1774 

25  1774 
20*  1774 

27  1775 

28  1775 

29  1775 

30  1775 

31  1775 

32  1775 

33  1776 

34  1776 

35  1776 

36  1777 


.1.  P.  Diwld. 
Abbu  Uozier. 


MEiNK^lKS    ON 

(l«!liiP.  <1(!  UoifTe.  OhHorvations  HiirlVxperlcnccMlc  |H*r('IJ«!rli(r. 
\Iiozi,r]  Jonrn.  de  P/i!/n.,  2,  374-H. 

G.  L.  Lefliige.  KutlcxioiiH  8tir  uiic  noiivclle  ('Xporicnce  dii  Ku- 
vtircnd  Pi^re  Ucrtier.  [Hnzier]  Journ.  de  Phyn., 
2,  378-81. 

J.  P  Diivhl  iuul  Fiilhers  VoiW,  nnd  Ik-ilicM*.    (Notice  of  their  ex- 
p(Miin«;iits).     {Uozitr]  .hmni.  dr  Phyn.,  4,  338. 
|{epoi)geuiix  rOrtoxioiiH  tic  M.  Leaiige.     [Roziei\ 
JoHvn.  de  P/iyx.,  4.  431-41. 
Observiilions   hut    l.i   Ictirt!   de    Pere    Bertier. 
[liozier]  Journ.  de  I*/iys.,  4,  454-61. 

Father  Bertier.  (Account  of  experiments).  Jonrn.  de  Verdun, 
148-185. 

.1.  P.  David.  Sur  la  pesanteur  des  corps.  {Rozier'\  Journ.  de 
Pkyit.,  5,  120-139. 

Fatlier  Bertier.  (Letter).     {R>zin]  Jonrn.  ile  Pfiyn.,  5,  305-18. 
—  (Account  of  cxp'lH.  tniMle  by  coin,  of  Acad,  of 

Dijon).     [liozier]  Journ.  de  P/iyn.,  5,  314-26. 

Chev.  de  Doloniieii.  Experiences  siir  la  pesantenr  des  corps  t\ 
differentes  distances  dn  centre  de  la  tene. 
[Rozier]  Journ.  de  P/iyn.,  l\,  1-5. 

N.  Maslielyne.  A  proposal  for  measuring  llie  attraction  of  some 
bill  in  this  king<lom  by  astronomical  observa- 
tions.    Phil.  Trann.  Jjond.,  495-9. 

N.  Maskelyne.  An  account  of  observations  made  on  the  moun- 
tain Schehallien  for  finding  its  attraction.  Phil. 
Trann.  Loud.,  500-42. 

F.  K.  Achard.  Bemerkung<!n  Uber  die  von  Herrn  Bertier  an- 
geslelllen  Versuclie.  lieschiift.  der  Berl.  Oes. 
Naturf.  Freunde,  2,  1-11. 
Experiences  et  vues  sur  I'intensite  de  la  pes- 
anteur dans  I'inlerieur  de  la  terre.  [Rozier] 
Journ.  de  Phyn.,  7,  1-12. 

Discours  sur  I'altraction  des  montagnes ;  traduit 
par  M.  le  Hoy.  [Rozier]  Journ.  de  Phys.,  7, 
418-34. 

Retraction  du  P^re  Bertier  de  I'Oratoire,  sur  la 
consequence  qu'il  a  tire  de  son  experience  d'un 
corps,  pesant  plus  dans  un  lieu  haut  que  dans 
un  bas.  [Rozier]  Journ.  de  Phys.,  9,  460-6. 
An  account  of  the  calculations  made  from  the 
survey  and  measures  taken  at  Schehallien  in 
order  to  ascertain  the  mean  density  of  the  earth. 
Phil.  Trans.  Lond.,  68,  689-788. 
Calculations  to  determine  at  what  point  in  the 
side  of  a  hill  its  attraction  will  be  the  greatest. 
Phil.  Trans.  Lond.,  1-14. 

Experiments  to  determine  the  density  of  the 
earth.     Phil.  Trans.  Land.,  88,  469-526. 
146 


Q.  L.  Lesage. 


J.  Pringle. 


Father  Bertier. 


37    1779    C.  Hutton 


38    1780    C.  Hutton. 


39    1798    H.  Cavendish. 


T  II  K    LAWS    ( )  K    ( J  U  A  \'  I  T  A  T  I  ( )  N 


89i  1799 
40    1799 


L.  W.  Gilbert. 


an- 

Oes. 


the 
I   in 

mb. 

the 

test. 

the 


41     1808    A.  Motte 


43    1806 


43  1809 

44  1810 

45  1811 


F 


P.  X.  von  Znch. 


C.  Hiittou. 


46    1811    J.  Play  fair 


47  1812 

48  1818 


49  1814 

50  1815 

51  1819  T.  Young 


52  1820 

53  1821 

54  1821 


(U.'vi«'w  of  80).  imi.  lint.,  11.  288-41. 

VtMHUclic,  iiin  (lie  Dichti^ktit  /.u  Ixstinitnfti, 
von  llenrv  Ciivnidlsli,  Ksq.  [GUlHi't]  Ann.  (Ur 
/»/<//«.,  2/1-02. 

The   niutlii  tnatieal   principles  of  natural    |)lii 
losopby  by  Sir  I.  Nuwton,  traiiHlateil  into  Knglisli 
Ity  Andrew  IMotte,  to  whicli  are  added  Newton's 
syHteni  of  the  world,  etc.      VV.    Davis'  e<l".    8 
vol.     London.     8'". 

H.  W.  Rranch'.s.  Theoretisclie  I'nter.sueliungen  nbi-rdieOseiila- 
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8U(!hen  Uber  die  Attraction  kleiner  MaNsen. 
Mag.  fiu'  den  Nt'iienten  ZKntand  tltr  Aalurkiindc, 
12,  800-810. 
X.  von  Zacli.  lJel»er  dit;  IM(\i;llchkeit  die  Oestalt  der  Krde 
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Corre>ip.,*li\\\A). 

Ueber  Densiidt  der  Krde  und  deren  EintUis.s 
anf  geographisclie  Ortsbestininuingen  MomiU, 
6V/v.<*/>,  21,298-810. 

On  the  ealenlations  for  ascertaining  the  meiin 
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112-6. 

Account  of  a  lithological  Hurvey  of  Schehallien. 
made  in  order  to  determine  the  .specitic  gravity 
of  the  rocks  which  compose  that  mountain. 
Phil.  TmuH.  Lond./Ml-n. 
Tracts  on  mathematieid  and  philosophicul  Nub- 
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Bericlit  von  einer  lithologischon  Aufnahme  des 
Schehallien,  um  das  specifiscrhe  Oewicht  der 
Gebirg.sarten  dessejhen,  und  daraus  die  mittlere 
Diehtigkeit  der  Erde  zu  be^tiinmen,  von  J.  Play- 
fair.  Esq.     Pof/f/  Atin.,  43.  62-75. 

F.  X.  von  Zach.  L'attraction  des  montagnes,  et  ses  effets  sur  les 
tils  il  plomb  ou  sur  les  niveaux  des  instruments 
d'ustroMomie.     2  vol.     Avignon.     4*°. 

N.  M.  Chompre.    E.xperiences  pour  determiner  la  denslte  de  la 
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glais.    Jonrn.  de  Vfjc  Hoy.  Polytechnique.    Cfthier 
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Remarks  on  the  prol)ahilities  of  error  in  physical 
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Phil.  Trans.  Lond.,  70-95  ;  Misc.'Workfi,  2,8-28. 
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On  the  mean  density  of  the  earth.  ITilloch] 
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(Same  title  as  53).     Phil.  Tranx.  Lond. ,  276-292. 
147 


C.  lIultoD. 
L.  W.  Gilbert 


C.  Hutton. 


C.  Hutton. 


C.  Hutton. 


MKMUIUS    ON 


5.")    IH'.M    F.  Curlinl. 


50  1825  8. 

57  1H20  M 

5H  18'^7  K 

Ml  IH'J?  M 


60     1827    — 
01     1825-45 
02*  1828    — 

68     1828     M. 


04    1829-30 

65    185}:}    S, 
00     185J7    K. 


07  1838  F.  Heicb. 

08  1838  — 


69  1839  F 

70  1840  C. 


71  1840  L. 

72  1840  A. 

73  1841  L. 

74  1842  J. 


OfiservHzlonI  drllii  Iiini?liP//,ii  drl  pon<l<ilo  sem- 
plicr  fiittt!  111!'  hII(>/./ii  (li  iiiilli;  Ichc  niiI  livolldtiel 
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VV.  Drobiflcii.    Do  vein  lunne  fl^^iiru.     liipsiiie.     12""'. 

S(iiliiiie).  All  tu;cotiiit  of  Pruf.  Ciuliiii'H  expcriinciits  on 
Moni  (;<'iiiH.     Quart.  Joiirn.  ofSe.,  24,  153-9. 

W.  DrobiHch.    I'cbei'dii^  in  dcii  Mincii  von  Dolcoatli  in  Ooiii- 
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.I.S.  T.  Oeblei.  Pliysiliiilisclies  WOiterbuch.  22  vol.  Leip- 
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Account  uf  e.\pci'iraent.s  inudc  ui  Dolcoatli  mine 
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bri«lge.     8*".     Printed  privuiely. 

\y.  Drobiscli.  AiisfUhrlielier  Ucrlcht  llbcr  inelirerc  in  den 
Jahren  1820  iind  1828  in  den  Minen  von  Dolcotitii 
io  Cornvvull  ziir  Bestiramung  der  niittleren  Dicli- 
tlgkeit  der  Knle  ungeHtellte  I'endelversucbe. 
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J.  C  E.  Scliinidt.  Jidirbiicli  der  mutliemiitisclien  und  pliy- 
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I).  PoissoD.     Traitede  iiu'c,iiii(pU'.    2'' ed".    2  vol.    F^aris.  8'". 

d(;  Beaumont.     Fxtruitd  un  incinolrc  de  M.  Ueicli  Hiir  la  dcii- 
site  dc  la  terrc     Cotitp.  Uend..  5,  097-700. 
Versuclie  ttbcr  die  iniitlcie  Diclitigkeit  der  Erde 
mittelsl  der  Drehwage.     Freiberg.     8*". 
On  the  repetition  of  the  Cavendish  experiment, 
for  determining  the  mean  density  of  the  earth. 
Phil.  Mag.,[^,  12,  283-4. 
(Same  title  aa  08).     Man.  Nut.  Roy.  Aatr.  iSoc, 
4,  90-7. 

Sur  la  determination  de  In  densite  moyoime  de 
la  terre,  deduite  de  I'observation  du  peminle  faite 
a  I'Hospice  du  Mont-Cenis  par  M.  Caiiini  en 
Septcmbre,  1821.  Mem.  Accad.  Torino,  [2],  2, 
379-84. 

F.  Menubrea.    Calcul  de  la  densite  de  la  terre.     Mem.  Accad. 

Torino.  [2],  2,  305-08. 

G.  Calcul  de  la  densite  de  la  terre,  par  L.  F.  Mena- 
brea.    Bibl.  Univ.  de  Geneve,  [nouv.],  27.  108-75. 

F.  Menabiea.    On  (Cavendish's  experiment.     Phil  }faf/.,  [3], 

19,  02-3. 
F.  Saigey.      Densite  du  globe.     Rev.  Scient.  etind.,  [Quesne- 

lille].  11.  149-60  and  242-53.  and  12.  373-88. 
148 


Baily. 
I.  Giulio. 


Til  K    LAWS    OK    <i  KA  VII'ATIoN 


[3], 

'fine- 
88. 


75  IH43     F.  Haily. 

76  IH42    F.  FtuHy. 


77     1842 


78  1H4;} 

79  1843 

80  184;{ 
80i  1845 


F.  Ilaily. 

F.  Bully. 
F.  Bully. 

A.  G. 

('.  A   F.  Pouts, 


81  1847  G.  W,  Ilearn. 

82  1849  E.  Sabine. 

88  185a  F.  Reich. 

84  1853  — 

85  1853  Scliaar. 

86  1853  — 


87  1853 

88  1855 

89  1855 

90  1855 


F.  R^ich. 

G.  B.  Airy. 
G.  B.  Airy. 


An  atM-otiMt  of  Honx-  <x|H>riini'iiiM  with  tlm  tor 
hIoii  rod,  for  ilctcriiiinlii^  tin-  mean  ilniMiiy  of 
the  •Mirlh.      /»/*//.  .\/,i;/  ,  |;{|,  21,  il  I   •,»!. 
Bi'Sllllats  lie  (|l|)-|<|ll)'H  c.XlttM  iriirf>  lulliH  iivcc  lit 

hulann'  ilc  iiiisiiiii,  |i(iiir  tU'-irniiini'r  In  dciiHiiC' 
niovcMiinlr  III  Irrro.      Ann.  ilr  Chiin.  tttlr  l*/ii/tt., 

|:<|!  rt.  :{;iH-5:{. 

licriclit  von   cinl^rii  Vcrsiichcti  mil  dcr  Dich- 

wa^e  /iir  ItiNtimmimgdtr  mlltlcnn  Dichligkelt 

dcr  Krde.     /•»/////.  Ann.,  57.  4:.:{  67. 

(Siiint!  litic  as  75).      .\t<>ii.  .\i<f    Uoi/.  Axtr.  .*v>r. , 

rt,  IHH  II  nd  197-'.»()6. 

KxpcriiiuMits  with  the  torsion  tod  ft)r  dctcrndn- 

iuu;  the  mean  diMisily  of  the  earth.     Mmi    li'ii/. 

Astr.  .*^/r  ,  14.  1-130  and  i.-eexlviii. 

(Same  liile  as 76).  liihl.  Unii\  ik  (ienhr,  [none], 

4:  J,  177-81. 

Von  <ien  kleinen  Ahlenkiin;;*  n  dcr  FiOthlinic 
(iiid  dcs  Niveaus,  welelic  diireli  die  An/iehuiip;- 
er>  tier  Sonne,  d«-s  Mondes,  initl  eiiiiu;rr  lern'S- 
trischt'ii  Grgmsiilndc  jn'ivorgeltniclil  wcKh-n. 
Autt:  X,irfi.,*2*2,  33-43. 

On  tlu!  cause  of  the  discrepancies  observed  by 
Mr.  Baily  with  the  (.'avendi^h  apparatus  tor  de 
teiinining  the  mean  density  of  the  earth.  Phil. 
TinnH.  Land.,  317-39 

Cosmos,   by  Alexander  von   llumitoldt,  trans- 
lated under  the  superintendence  of  Lieut. Col. 
Edward  Sabine.     6"' Ed".      1 .  liondon.     H*". 
Neue  Versuche  nni  der  Di'ehwaage.    Jj<ip.  Abh. 
math.  phy.  cL,  1.  383-430. 

Neue  Ver.HUche  rd)er  die  ndlllere  Dielititrkeit  der 
Erde.  von  F.  Reich.  /V/.V-  Ami-,  8*>.  lHO-98. 
Rapport  de  M.  Schaar  sur  un  nienutire  de  M. 
Montigny  relalif  aux  experiences  pour  deter- 
miner la  densite  tie  la  terre.  hull.  Acad.  Ron, 
Belg.,  19.  pt.  3,  476-81. 

Nonvelles  experiences  sur  la  densite  moyenne 
de  la  lerre.  Ann.  de  Chim.  et  de  P/iyn.,  [3],  HH, 
383-3. 

New  experiments  on  the  mean  density  of  the 
earth.     P/iil.  Mof/.,  [4],  5,  1.54-9. 
(Report  on    lliirton  expts.).      Mon.  Not.  Roy., 
Astr.  Soc.,  15,35-6. 

Noterespeciing  the  recent  experiments  in  the 
Harinn  Colliery.  Mon.  Not.  Roy.  Antr.  Soc, 
15,  46. 

(Report  on    Harton  expts.).      Mon.  Not.  Roy, 
Astr.  Soc.,  16,  135-6. 
149 


MKMOIIIS    ON 


Ft> 


91    1855    — 


93    1855    — 


93    1855    J.  H.  Pratt. 


94     1855    G.  B.  Airy. 


95  1855 

96  1856 

97  1856 

98  1856 

99  ,856 


T.  Young. 
J.  H.  Pnitt. 
G    B.  Airy. 

J.  H   Pratt. 


100    1856    G.  B.  Airy. 


101    1856    G.  B.  Airy. 


103    1856    G.  G.  Stokes. 


103    1856    H.  James  and 


Nolo  sur  Irs  olwervations  dii  pendule  executet-s 
dans  Ics  inincs  d«!  Marlon  |)our  deierminer  la 
dcMisiie  m<»y(!nn('d(!la  terre  :  par  M.  Airy.  Ann. 
de  C/iim.  H  dc  PIiiih.,  [3],  4,*$,  381-3. 
Exlrait  (III  rapp«  rt  preseiiit'  tl  la  35"'«  seance 
annivcr-saire  de  laSocieti'  H^yale  Astrononiiqiie 
<Je  Londies  par  \v.  conseil  (Ic  ccile  societ'';  le  9 
Fevricr,  1855.  Arch,  dex  Sc.  Phj/n.  et  Nat.,  liJ>, 
188-191. 

On  the  atinietion  of  the  Himalaya  mountains, 
and  of  the  elevaled  rcgioris  beyond  them,  upon 
the  plumb-line  in  India.  Phil.  Trans.  Ijond., 
145,53-100. 

On  the  eDUiputation  of  the  effect  of  the  attrac- 
tion of  njounlain-masscs.  a.s  disturbini?  the  ap- 
parent astronondcal  latitude  of  siations  in 
ffeodetic  surveys.  Phil.  Trans.  Loud.,  14l>, 
101-4. 

Miscellaneous  works  and  lift^  by  Peacock  and 
Leitch.     4  vol.     London.     8'''. 
(Same  iltle  as  93).     Man.  Not.  lioy.  Astr.  Soc, 
Hi,  36-41  and  104-5. 

(Sa'ue  title  as  94).  Mon.  Not.  Roy.  Astr.  Soc, 
lO.  42-43. 

(Report  on  Ha''ton  expts.).  Mon.  Not.  Hoy. 
Astr.  Hoc,  16,  i04. 

On  the  effect  of  local  attraction  upon  the  plumb- 
line  r-l  stations  on  the  Euirli^h  arc  of  the  merid- 
ian, between  Dunnose  and  Burleigh  Moor  ;  and 
a  meihi'd  of  computing  its  amount.  Phil.  Trans. 
jMnd.,  14«.  31-5'>. 

Account  of  p(!ndi;lum  cxperimenis  umlertaken 
in  th(!  Ilarton  Colliery,  for  the  purpose  of  de- 
termining the  mean  density  of  theearih.  Phil. 
Trans.  Lond.,  146,  997-342. 
Supplement  to  the  "account  of  pendulum  ex- 
perinients  undertaken  ia  the  Harlon  Colliery"  ; 
lieing  an  account  of  experiments  undertaken 
to  determine  the  correclioii  for  the  temperatiu'c 
of  the  pendulum.  Phil.  Trans.  Lond.,  146, 
343-55. 

(Addendum  to  101  ;  on  the  effect  of  the  earth's 
rotation  and  ellipticity  in  modifying  the  numer- 
ical results  of  th<;  Harton  experiment).  Phil. 
Trans.  Lond.,  146,  353-5. 
A.  R.  Clarke.  On  the  deflection  of  the  plumb- 
line  at  Arthur's  Seal,  and  tl.j  mean  specitie 
(gravity  oi  the  earth.  Phil.  Trans.  Lond.,  146, 
591-6CG. 

150 


THE    LAWS    OF    GRAVITATION 


104     1850    II.  James. 


105    1856 


106    1856    S.  Haughton. 


107  1856    G.  B.  Airy. 

108  1856    H.  Jumes. 


109  1856    G.  B.  Airy. 

110  1856    — 


111     1856    G.  B.  Airy. 


113    1857    E.  R. 


113    1857    — 


114  1857  — 

115  1857  — 

116  1857  — 

117  1857  H.  James. 

118  1857  W.  ^.  Jacob. 

119  1857  G.  B.  Airy. 

120  1857  H.James. 


On  the  flgur,",  dimension.s  and  mean  specific 
gravity  of  llie  earlli,  as  derived  from  the  ord- 
nance trigoiKiinetrical  survey  of  Great  Britain 
and  Ireiaiui.  Pliil.  Traus.  Land.,  I4«,  607-26. 
Ucher  dii:  in  der  Iv«)ideiigrnl)e  von  llarton  /,nr 
Bestiniiniiiig  der  tnittleren  Di(;lile  der  £rde  nn- 
ternonimenen  Pendelbeohaciilungen  ;  von  G.  B. 
Ai ry .  Pogg.  Ann.,\)7,  599-605. 
On  tlie  density  of  tlie  jiirili,  deduced  fronfl  tlie 
experiments  of  the  Astronomer  Iloyal,  in  the 
Harton  coal-pit.  Phil.  Mag.,  [4],  12,  50-1. 
(Same  title  as  100).  Phil.  May.,  [4J,  111.  226-31. 
Account  of  the  observations  and  computiU^ons 
made  for  the  purpose  of  ascertaining  the  amount 
of  the  deflection  of  the  plumb-line  at  Arthur's 
Seat,  and  the  mean  specific  gravity  of  the  earth. 
Phil.  Mag.,  [4],  12,314-6. 
(Same  title  as  101).  Phil.  Mag.,  [4].  12,  467-8. 
Ueber  die  Dichtigkeit  der  Erde,  hergeleitet  aus 
den  Verauchen  des  Kttnigl.  Astronoraen  (Hrn. 
Airy)  in  der  Kohlengrube  Harton  ;  von  Sr.  Ehr- 
wtlrd.  Samuel  Haughton,  Fellow  des  Trinity 
College  in  Dublin.  Pogg.  Ann.,  »9,  332-4. 
On  the  pendulum  experiments  lately  made  in 
the  Harton  Colliery,  for  ascertaining  the  mean 
densitv  of  the  earth.  Am.  Journ.  Sc.,  [2],  21, 
359-64. 

Memoire  sur  lea  experiences  enterprises  dans  la 
mine   de    Harton    pour   determiner   la   densite 
moyenne  de  la  terro,  par  G.  B.  Airy.     Arch,  des 
Sc.'Phi/.'i.  et  Nat.,  35,  15-29. 
Ueber  die  Diciitigkeit  der  Erdc,  hergeleitet  aus 
den  Pendelbeobachtungen  des  Herrn   Airy  in 
der  Kohlengrube  Harton  von  Herrn  S.  Haugh- 
ton, Fellow  am  Trinity-Collfge  in  Dublin.   Zeit. 
filr  Math.  u.  Phya.,  2.  68-70. 
Ueber  die  Bestimmungder  nnttleren  Dichtigkeit 
der  Erde.     Zeit.  filr  Math.  u.  Phys..  2,  128-30. 
(Same  title  as  103).     Proc.  Boy.  Soc.  Edin.,  ii, 
364-6. 

(Notice  of  106).  Am.  Journ.  Sc.,  [2],  24,  158. 
(Same  title  as  104).  Phil.  Mag. ,  [4],  1 3, 129-32. 
On  the  causes  of  the  great  variation  among  the 
different  measures  of  the  earth's  mean  density. 
Phil.  Mag.,  [4],  13.  525-8. 
(Same  title  as  101).  Proc.  Roy.  Soc.  Land.,  8, 
58-9. 

(Same  title  as  104).     Proc.  Boy.  Soc.  Lond.,  8, 
111-6. 

151 


MEMOIRS    ON 


131 

1857 

W.  8.  Jacob. 

122 
123 

1858 
1858 

G.  B.  Airy. 

124 

1858 

— 

125 

1858 

H.  James  and 

Proe.  Roy.  Soc.  Lnrm.,  8, 


Proc.  Hoy.  Ind.,2,  17-22. 
Man.  Not.  Roy.  Astr.  Sac, 

Moil.  Not.  Roy.  Adr.  Soc, 

r 


126 
127 

128* 


1859 
1859 
1859- 


P.  F. 


60 


129    1861    O.  Struve 


130 


131 


1863 
1864 


137    1873 


(Same  title  as  118). 

295-9. 

(Sume  title  as  111). 

(aame  title  as  103). 

18,220. 

(Siimc  litlc  as  104). 

18,  220-2. 
A.  It.  Clarke.    Ordnance  trigonometrical  Survey 

of  Great  Britain  and  Ireland,     Account  of  llie 

observiitions  and  calcniations  of  the  principal 

triangul.-ition  ;  and  of  the  figure,  dimensions  and 

mean   specific  gravity  of  the  earth   as  derived 

therefrom.     2  vol.     London.     4'°. 

(Same  title  as  125).     Mon.  Not.  Roy.  Astr.  Soc, 

1»,  194-9. 
J.  Gosselin.  Nouvel  examen  sur  la  densite  moyenne  de  la 

terre.     Mem.  Ac<ul.  Imp.  de  Metz,  [2],  7,  469-85. 
E.  Sergent.     Sidla  densita  della  materia   nell'  intorno  del 

globo,  e  suila  polenza  della  crosta  terrestre.    Atti 

della  Soc.  lUil.  di  Se.  Nat.  Milano.  2,  169-175. 

Ueber  eiiien  von  General  Schubert  an  die  Aka- 

demie  gerichteten  Antrag  betreflFend  die  Rus- 

sisch  •  Scandinavische  Meridian  -  Gradmessung. 

Bull.  Acad.  St.  Petersb.  phys.  math,  cl.,  3,  395- 

424. 
H.  A.  E,  A.  Faye.  Sur  les  instruments  geodesiques  el  sur  la 

densite  moyenne  de  la  terre.    Comp.  Rend.,  56, 

557-66. 
E.  Peclimann.  Die    Abweichung   der  Lothlinie   bei  astrono- 

mischen   Beobachtungsstationeii   und    ilire  Be- 

rechnung  als  Erforderniss  einer  Gradmessung. 

Denkftchr.  Acad.  Wm.  Wien.  math.-naturw.  cl., 

22,  41-88. 

Note  surle  calcul  de  I'experiencede  Cavendish, 

relative  A  la  masse  el  fi,  la  densite  moyenne  de  la 

terre.     Cosmos,  24,  543-5. 

A  treatise  on  attractions,  Laplace's  functions, 

and  the  figure  of  the  earth.   '6^.  Ed".    Cambridge 

and  London.     8*". 

Ueber  die  nnltlere  Dichtigkeit  der  Erde.    Zeit. 

fur  Math.  n.  Phys.,  lO,  224-7. 

Ueber  die  Hestimniungder  millleren  Dichtigkeit 

der  Erde.    G5ttingen.     4'°. 

Sur  le  calcul  de  la  densite  moyenne  de  la  terre, 

d'apr^s  les  observations  d'Airy.  Ball.  Acad.  Roy. 

Belg.,  [2],  33,  369-372  and  389-409. 
A.  Cornu  et  J,  B.  Bailie.    Determination  nouvelle  de  la  con- 

stante  de  I'attraction  et  de  la  densitfe  moyenne  de 

la  terre.     Comjt.  Rend.,  7<»,  954-8. 
152 


132 

1864 

J.  Babinet. 

133 

1865 

J.  H.  Pratt. 

134 

1865 

H.  Scheffler. 

135 

1869 

A.  Scbell. 

136 

1872 

F.  Folic. 

T  II  E    L  A  W  S    O  K    0  R  A  \'  I  T  A  T  I  ()  N 


la 


.'it 


138*1873    —  (Notice  of  137).    Bull.  Ilebd.  de  l\Umt\  Scient.  (fr 

France,  [IJ.  12,  70. 

139  1873    A.  Coinu  and  J.  H.  Hiiille.   Miitniii  (Ictcnniniiiion  of  \\w  cnn 

stiuit  of  iiUriictioii,  and  of  the  mcim  density  ol 
tlic  eaiMi.     Chemifdl  Nrwx,  27,  211. 

140  1873    I.  Todliunler.  A  history  of  Wm  iimiliemfUicil  theories  of  at 

tractirtn  iitid  the  fiijiire  of  the  earth,  from  the 
time  of  Newton  tothatof  Laplace.  2  vol.  Lon- 
don.    8*". 

141  1878    A.  Cornii  et  J.  B.  Bailie,     ^^tnde  de  la  renistance  de  I'air  dans 

la  balance  de  torsion.     C'omp.  lieitd.,  8($,  571-4. 

142  1878    A.  Cornii  et.I.  li.  Bailie.     Sur  la  mesiire  de  la  densite  moyenne 

de  la  terre.     Cnnp.  Jiaid.,  8«,  61)9-702. 
148     1878     A.  Cornu  et  J.  B.  Bailie.   Iidliieiice  des  terrnes  proportionels  an 

cane  des  ecarts,  dans  U'  nioiivemt-nl  oscillatoin; 
do  la  balance  de  torsion.  Comp.  liend.,  80, 
1001-4. 

144  1878     Ph.  von  Jolly.  Die  Anwendiini;  der  Waagc  auf  Probleme  der 

Gravitation.  Part  1.  Abh.  Bay.  Aknd.  Wisn.cl. 
2.  1»,  Ahth.  1.  157-176. 

145  1878    Ph.  von  Jolly.  (Same  title  as  144).     117^^.  ^mh..,  5,  112-34. 

146  1879    J.  IL  Poynting.  On  a  method  of  using  the  Indance  with  gnat 

delicacy,  and  on  its  employment  to  determine 
the  mean  density  of  the  earth.     Pr<>c.  Rotj.  JSoc. 
Land.,  28,  2-35. 
H.  A.  E.  A.  Faye.     Sur  les  variations   seculaires  de  la  figure 
mathematique  de  la  terre.     Comp.  Rend.,  OO, 
1185-91. 
Faye.    Sur  la  reduction  des  observations  du  pen- 
dule  au  niveau  de  la  mer.     Comp.  Rend.,  90, 
1443-6. 
F.  R.  Helmert.     Die   mathen^atischen    tind   physikalischen 
Theorieen  der  hOlierei?  jreodasie.     2  vol.    Leip- 
zig.    8*°. 
O.  Zanolti-Bianco.     U  problema  meccanico  della  figuradella 
terra.     2  parts.     Firenze  Torino-Roma.     8'°, 
A.  R.  Clarke.   Geodesy.     Oxford.     8^". 
O.  Knopf.    Ueber  die  Methoden  zur  Bestimmung  der  mitlleren 

Dichtigkeit  der  Erde.    Jena. 
T.  C.  Mendenhall.  Determination  of  the  acceleration  due  to  the 
force  of  gravity,  at  Tokio,  Japan.     Am.  Jonni. 
i,  Se.,  [3],  20,  124-32. 

T.  C.  Mendenhall.  On  a  determination  of  the  force  of  gravity 
atlhe  summit  of  Fujiyama,  Japan.     Am.  Journ. 
Sc.,  [3J.  21,99-103. 
153    1881    F.  Keller.         Sulla   diminuzione    della   gravita   coiraltezza. 

Atti  Accnd.  TAncei.  Mem.  d.  sc,  [3].  J),  103-17. 
153    1881     Ph.  von  Jolly.  (Same  title  as  144).     Part  2.     Ahh.  Bay.  Akad. 

WisHcl.  2.  14.  Ahth.  2.  3-26. 
153 


146i  1880 


147     1880    H.  A.  E.  A 


148    1880-4 


148i  1880-5 

149  1880 
149^*  1880 

150  H80 


151    1881 


MEMOmS    ON 


M: 


•I 

1 


154     1881 
1544  188a 


155  1883 

156  1883 

157  1883 

158  1884 

159  1884 

160  1885 

161  1885 

162  1885 

163  1885 

164  1885 

165  1886 

166  1886 

167  1886    F.Keller. 

168  1887    F.  Keller. 

169  1887    J.  Wilsing. 

170  1887    J.  Wilsing 


Ph.  von  Jolly.  (Siinie  title  as  153).     Wied.  Ann.,  14.  331-55. 

J.  G.  Walleiilin.  UelxT  <lie  Metluxlen  zur  Bt'stinunung  ilcr 
initlleren  Diclilc  dcr  Erde  iiiid  eine  neue  dies- 
hezttglicliu  Anvveiidung  der  Wage.  Humboldt, 
1,  312-7. 

R.  voij  Steineck.  Unterauclmngeri  ttber  die  Schwere  itn  In- 
nern  der  Erde.  Mitth.  Mil.-Oeog.  Inst.  Wieiii,  2, 
77-130. 

R.  von  Sterneck.  Wiederlioliing  der  Untersucliungen  Uber  die 
Schwere  im  Iniiern  tier  Erde.  Mitth.  Mil.-Qeog. 
fust.  Wien,  ',i,  59-94. 

J.  B.  Bailie.  Sur  la  resistance  de  I'air  dans  les  niouvements 
oscillatoires  trt's  Icnts.  Coinp.  Rend. ,  OO,  1493-5. 

R.  von  Sterneck,  Unlersnchniigen  liber  die  Schwere  aiif  der 
Erde.     Mitth.  Mil.-Geog.  Inst.  Wien,  4,  89-155. 

A.  KOnigand  F.  Richarz.  Eine  neue  Metliode  zur  Besiininiung 
d(;r  Gravilationsconstante.  Sitzunysh.  A/cad. 
Wiss.  Berlin,  1303-5. 

A.  Kttnig  and  F.  Riciiarz.  (Same  title  as  159).  Wied.  Ann., 
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A.  M.  Mayer.  Methods  of  determining  the  density  of  the  earth. 
Nature,  31.  408-9. 

A.  KOnig  and  F.  Richarz.  Remarks  on  our  method  of  deter- 
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J.  Wilsing.  Ueberdie  Anvvendung  des  PendeKs  zur  Bestim- 
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R.  von  Sterneck.     Fortscizung  der  Untersuchungen  Uber  die 
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R.  von  Sterneck.  (Same  title  as  155).  Mitth.  Mil.-Oeog.  Inst. 
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W.  M  Hicks.  On  some  irregularities  in  the  values  of  the 
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Sulla  deviazioiie  del  tilo  a  piombo  prodotta  dal 
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Bestimmung  der  mittleren  Dichtigkeit  der  Erde 
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164 


THE    LAWS    OF    (JUAVITATION 


171    1888    J.  H.  Gore. 


lie 


173 

1889    J.  VVilsing. 

173 

1889    VV.  Laska. 

174 

1889    J.  H.  Gore. 

175 

1889    C.  V.  Boys. 

176 

1889-90    (;.  V.  Br 

177 

1889-90    J.  Joly. 

178    1889-91 


179    1890    Tliieseii. 


180    1891    J 


181     1893    A.  Berget. 


183 
183 


185 
186 


1893 
1893 


184    1894 


1894 
1894 


187 
188 


1894 
1894 


Deterininaiioii  of  the  mean  density  of  the  earlli 

l»y  ineiin.s  of  ii  pendulum   principle,  by  J.  Wil- 

sin^,    trtmsltUed   and   condensed.     Sniillisoniiin 

Uep.  1H88.     635-46. 

(Same  111  leas  170).     PiiM.  Axtropfiyn.  Ohn.  Pntn 

(fain,  «,  Si  lick  3,  133-91. 

Ueher  eincn  neuen   Appiirat  zur  Be.slimmung 

der    Erddichte.      Zeit.   J'iir   Ind.-  Kundc,    1>, 

354-5. 

A  bibliogniphy  of  geodesy.     Wasliingtnu.     4'". 

A  pp.  to  U.  S.  Coast  and  Geod.  Surv.  Hep.  for 

1887. 

On  the  (.'avendisli  Experiment.     Proc.  Hot/.  Soc. 

/.<>//</.,  40,253-68. 
(;.  V.  Boys.     (Same  title  as  175).     Nature,  41.  155-9. 

(Report   of   meeting   of    Univ.  Exptl.    Assoc. 

Dublin).     Nature,  41.  256. 

Collection  de  memoires  relatifs  i\  la   physique, 

piiblies  par  la  Societe  Franyaisc  de  Physique, 

4  and  6.     Paris.     8'°. 

Determinati<)n  de  la  variation  de  la  pesanteur 

avec  la  hauteur.   7'w/\  et  Mem.  dii  liur.  Internal. 

(lea  PoUIh  et  Men.,  7,  3-32. 
H.  Poynting.     On  a  determinaiion  of  the  mean  density  of 

the  earth  and  the  gravitation  constant  by  means 

of  the  cominnn   balance.     Pftil.   TratiH.  Land., 

[A],  182.565-656. 

Determination  experimcntale  de  la  constantede 

I'atlraction  universelle,  ainsi  que  de  la  masse  et 

de  la  densite  de  la  terre.     C'onip.  Rend.,  116, 

1501  3. 

Sur  la  realisation  des  temperatures  constanles. 

Comp.  Rend.,  117,  96-7. 
F.  Richarz  und  O.  KriuaiMeuzel.     Die  Abnahnie  der  Schwere 

niit  d«r  I  lolie  l»<si  immt  dureli  Wilgungen.    Sitz- 

unffsh.  A/>(i»f.   Wixtt.  Berlin,  163-83. 
F.  Richarz  und  O.  Krigar-Menzel.     (Same  title  as  183).      Wied. 

Ann.,  51,  559-83. 
J.  H.  Poynting.     The  mean  density  of  the  earth.    London.  8'". 
J.  H.  Poynting.     A  hisiciy  of  the  methods  of  weighing  tln^ 

earth.     Proe.  lUrniinfjh<(ni  Nat.  Hist,  and  Phil. 

Soc,  9,  1-23. 

Die  Methoden   zur  Bcstimmung  der  mittleren 

Dichte  der  Erde.    Wm.  Beila(/e  zuni,  Jahresb.  des 

Oym.  zu  Weissenburr/  i.  ElmxK. 

On  the  Newtonian  consiantof  gravitation.  Proc. 

Roy.  Soc.  Lond.,  56,  131-2. 

(Same  title  as  187).     Nature,  50,  330-4,  366-8, 

417-9  and  571. 
155 


Gouy. 


186i  1894    G.  Fresdorf 


C.  V.  Boys. 
C.  V.  Boys. 


MKMOIKS    ON    TIIK    LAWS    OF    GRAVITATION 


II 


19:5     1896    C.  Brauu. 

194  1896-7    — 

195  1897    P.  Dnidc. 


189  1895    C.  V.  Bovs.      (Same  title  as  187).      Phil.  TmuH.  Lond.,  [A], 

18«,  1-72. 

190  1895    A.  S.  Mackenzie.    On  the  attractionsof  crystalline  and  isotropic 

masses  at  small  distances.     /*////.  Jicv.,  *2,  331- 
48. 

191  1896    F.  Kicharz  und  ().  Kii^ar-Menzel.     Gravitationsconstante  «nd 

mittlere  Dichtigkeit  der  P^nle,  hestitnml  durcli 
Wilj^iingcn.     Sitzin>f/nh.    A/cad.     Wixx.     Berlin, 
1305-18. 
I!)2     1896    R.  von  E5tv5s.     Unlersuclmnircn  [Wwv  Gravitation   und  Erd- 

ma/L^nielisniiis.      \Vi<'</.  Ann.,  51>,  354-400. 
Di(!  Graviiiitiotisconstanle,  die  Ma.ssc  und  mitt- 
lere Diclite  d(!r  Enle  nacli  ciner  neuen  experi- 
meniellen  Hcstimmung.     Denkxehr.  Ahid.  Wi»H. 
Wien.  niath.-nalnvw.  d.,  «4,  187-258c. 
The  gravitation  constant  and  the  mean  density 
of  tlu!  eartii.     Nature,  55,  296. 
Uebcr  Fernowirkungcn.     Wied.  Ann.,  62,  i.- 
xli.x. 

196  1897     L.  W.  Austin  and  ('.  H.  Thwing.     An  experimentiil  research  on 

gravitational  permeability.      IVn/.  Rev.,  5,  294- 
300. 
196i   1897-9    S.  Gllnther.  llandbucli  der  Geophysik.     2  vol.     Stuttgart. 

197  1897      J.  H.  P(oynling).     A  new  determination   of   the  gravitation 

constant,  and  the  mean  density  of  the  earth. 
Nature,  56,   127-8. 

198  1898    F.  Richarz  und  O.  Krigar-Menzcl.     Ucstimmung  der  Gravita- 

tionsconslantc!  und  mittlercn  Diclitigkeit  der 
Erde  dunth  Wilunngen.  Anhanr/  Ahh.  Aknd. 
WiHH.  Berlin.  1-196. 

199  1898    F.  Richarz  und  O.  Krigar-Mcnzd.     (Same  title  as  191).     Wied. 

^l//«.,  66,  177-193. 

200  1899    J.  H.  Foynting  and  P.  L.  Gray.     An  experiment  in  sean  h  of 

a  directive  action  of  one  quartz  crystal  on  an- 
other.    Phil.  Trans.  Lond.,  [A],  102,  245-56. 


156 


INDEX 


Acimrd,  49. 

Airy.  5,  IOC).  li:{,  118.  119.  121-124. 
128-130;  Thcoiy  of  Cuvcndish  Ex 
perinieiit,  100,  118;    Dolcoiith  Ex- 
periments.   113;    lliirton    Expcri 
niciits,  121. 

Arthur's  Bent.  118,  123.  124. 

Altraclion,  Newton's  Theorems  on. 
9  ;  Newton's  Error  in  Ciilculutioii 
of,  10,  17  ;  Primitive,  27  ;  Of  a 
Pliiteuu.  29-32  ;  Of  a  Spliericul 
Segment,  ('iilciiliiled  by  Newton, 
17  ;  l)y  (.'iirliiii,  Sclimidt  and  Giii- 
lio.  111.  112  ;  Shown  by  I)etle(;lion 
of  Pliiinl)-line.  33-43  ;  Of  Ciiiml)0- 
razo.  34.  39  ;  Of  Soiiehallien,  43. 
53-50  ;  Due  to  Tides,  44.  134  ;  Of 
uiiy  Hill.  (;aiculated  by  Iluiton, 
54;  Of  the  Great  Pyramid.  55; 
Local.  50,  122-124.  120,  134,  135, 
141  ;  Of  Mount  Mimet,  50  ;  Of 
Mass  Benealli  Earth's  Surface,  50, 

122.  123;   Of  Arthur's  Seat,  118. 

123,  124  ;  Of  t\M\x,  118  ;  Of  a 
Cone,  128 ;  Of  an  Infinite  Plane, 
135. 

Austin  and  Thwing,  142. 

B 

Babiiiet,  100. 

Bacon,  1,  2.  5.  49,  113 

Baily,  100,  105,  100.  115-120,  125, 
131-133.  137  ;  Cavendish  Experi- 
ment Criticized  by,  105  ;  Error  of, 
Pointed  out  6y  Cornu  and  Bailie, 
119  ;  x\nomalies  in  Results  of,  and 
their  Explanations,  118,  119. 

Balance,  Experiments  with  Beam. 
2-5,  48.  49.  125.  132.  140  ;  Experi- 
ments with  Torsion.  59-105.  114- 
121,  124,  135,  137-139.  142;  Mich- 
ell  Devised  Torsion.  60;  Experi- 
ments with  Pendulum,  131,  132. 


Bauernfeind.  124. 

lieaiimont.  110. 

Beruet,  124,  134,  135. 

Bertier.  47-49. 

Boscoviteh,  134. 

Bougiier.  5,  21 ,  23-25,  27,  32.  33,  36. 
39-44,  47.  53,  50.  130,  134;  On 
Tides,  44,  134  ;  Life  of,  44  ;  First 
to  Take  Account  of  Buoyancy  of 
Air.  26. 

Boyle.  4. 

Boys,  100.  135-137.  139,  142. 

Brandes,  105.  100,  120;  Theory  of 
Cavemlisli  Experiment.  106;  The- 
ory of  Oscillation  Method,  105, 
100,  120. 

BrauM,  100.  138.  139. 


C 


Carlini.  111-113,  128. 

Cavendish.  54,  55.  59.  90.  91.  98.  100, 

105-107.  114-110, 118,  119, 125. 135. 

130,  139;  Error  in  Calculation  of, 

100,  105  ;  Life  of.  107. 
Chimborazo,  22,  34.  39-41.  43. 
Clarke,  56.  118.  123,  124. 
Condamiiie,  de  la.  21.  28,  32.  36.  39- 

41.  43.  44  ;  Pendulum  Experiments 

of,  28  ;  Method  of,  for  Doubling 

Deflection  of  Plumb-line,  36. 
Cornu  and  Bailie.  66,  106.  115.  119. 

124.  125,  131.  135. 
Cotte,  48. 
Cotton,  4. 
Coulomb  Balance,  First  Proposed  by 

Michell,  60. 
Coultaud,  47,  48,  111. 


D 


D'Alembert.  31.  47. 
Damping.    Method    of    Finding    A, 
138,  141,  142  ;  Effect  of,  139. 


157 


INDKX 


!»•:, 


Ocvid.  47-49 

Dftleclion.  of  Arm  of  Torsion  H:il 
uiice,  How  McHsiircd,  by  (Javcn- 
(lish.  04.  98;  by  U.icli.  il6.  119; 
l)y  Biiilv,  117.  119.  i:{2.  133;  by 
liriiun.  138;  AlTittt.s  ihc  Period, 
97;  Error  in  iJaily's  Method  of 
()bs<'rviiiir.  119,  l'i5  ;  Mulliplied 
by  Poyntin^r.  133.  133. 

DcHcjirtes.  2.  49  ;  Sufjgestj'd  Method 
of  McHsuring  Griiviiy,  2 

Diiiu'iisionH  of  Torsion  Bidimce,  Ef- 
fVcts  of,  125.  135,  137,  138. 

Doleoath,  113,  121. 

Doloinien.  49. 

Drobisch.  113,  114. 

Dnidc,  142. 

E 

EOtviis.  106,  137,  138. 

F 

F)iye*31.  124,  130.  189;  Compensa- 
lion  Theory  of.  ('oriection  of  "  Dr. 
Young's  Rule,"  31,  124. 

Ferrel,  130. 

Flotation  Theory.  31.  124. 

Folic,  123 

F(.rl.es,  117,  118,  120. 

Forced  Vibrations,  138, 141,  142. 

Fr(?sdorf.  56.  11 3,  116.  123,  127.  128, 
131,  132,  135. 

Fujivuma,  127,  128. 


G 


Gilbert,  Dr..  1,  5.49. 

Gilbert,  L.  W.,  105. 

Giulio.  112. 

Gore,  124,  132. 

Gossflin,  106. 

Gouy.  135. 

Gnivinieter,  135. 

Gravitation.  Early  Conceptions  of, 
1,  49.  56  ;  Early  Experiments  on, 
by  Menibers  of  Royal  Society,  2-5  ; 
As  Explanation  of  Planetary  Mo- 
tion, bv  Newton,  2,  10-19  :  Msig- 
net.io  Theory  of.  1, 4. 5, 12  ;  liooke's 
Ideas  Concerning,  5,  6;  Compen- 
sator. 138 ;  Mnltiplicator,  138  ;  Per- 
meability, 142  ;  Velocity  of  Prop- 
agation i>f.  142. 

Gravity,  Proposed  Experiment  on, 
by    Bacon,    1  ;   by   Descartes,  2 ; 


Decrease  of.  with  Height.  27-83, 
47-49.  111-113,  126-128,  180,  187. 
140 ;  Law  of  Increase  of,  with 
Deptli,  129,  130  ;  Increase  of,  with 
Temperature,  131  ;  Mathematical 
DLscussHni  of,  frt>m  Potential,  137. 

Gray,  142. 

GUuther,  181.  141. 


Harton  Colliery.  5,  121,122 
Hanghton,  122. 
Hearri,  118,  120. 
Hebnert,  54,  56,  124,  127,  129. 
Hicks.  119,  131. 
Hooke,  2,  4,  5. 
Horizontal  variometer,  137. 
Humboldt,  56. 

Hutton.  54,  55,90,  100,  105, 106,  118, 
124. 

J 
Jacob,  56,  123. 

.binie.s  and  Clarke,  56,  118, 128, 124. 
.Icily,  125, 126,  187,  140. 
Joly,  138,  141. 

K 

Keller,  124.  127,  135,  137,  140. 
Kepler,  1.  2.  49. 
Kn«»pf.  113,  122. 
KOnig.  140. 

Krigar-Menzel.  140.  141. 
KrUmmungsvariometer,  137,  138. 


Lalande,  48. 

Laska.  141. 

Law.  of  the  Distance,  2.  9.  29,  47. 

101.  126.  142  ;  Of  the  Masses,  18. 

32 ;   Of  the  Material,  12,  142  ;  Of 

the  Medium,  142. 
Lesage,  2,  48,  49. 


M 


Mackenzie,  142. 

Maun(!tism.  Gilbert's  Explanation  of 
Gravitation  by.  1.  4.  5  ;  Contrasted 
with  Gravitation,  12;  Testa  for 
Eflfecis  of.  by  (Cavendish,  67.  68, 
75.  76  ;  by  Reich.  116,  120  ;  Sug- 
gested by  Hearn  to  Account  for 
Anomalies  in  Baily's  Results,  118- 
120. 


158 


INDEX 


MaHkelyne.  17,  48,  53-50,  101,  100. 

Miiyer,  140. 

Meimbreii,  55.  00,  100. 

MriMiunliiill,  124,  127,  128. 

Mcrcier,  47,48,  111. 

Micliell,  59,  GO,  01. 

Minu   Experiments,    1,   2,  4,  5,  40, 

113.  121,  128,  I'Jl. 
iMontigny,  119. 
Miiiicke,  55,  105,  100. 

N 

Nc'wion,  2,  0.  7.  9.  14-17.  19.  39.  43. 
47-49,  50.  107,  110,  124,  120.  141  ; 
ExpliiiDition  of  PliiiK-taiy  .Mmioiis 
by,  2,  10;  Pciidiiliim  Kxpcriiiujiit.s 
of,  11.  15  ;  Guess  as  lo  V^ahie  of 
A  by,  14  ;  Errors  in  ('iileiiliiiioiis 
of,  10.  17  ;  Indicates  Metlioils  of 
Finding  A,  17;  Calculates  Attnic 
tion  oi  u  Mouniain,  17  ;  Lite  of, 
19;  Attempts  to  Upset  Tlieorv  of, 
47. 

P 

Peclnnann,  124. 

Pendulum,  Experiment  with.  Pro- 
posed by  Bacon,  1  ;  by  Hooke,  5  ; 
Experiments  wi  ii,  by  Newion. 
10,  11,  15;  by  liouguer,  24-33; 
by  Coidlaud  and  Mercier.  47  ;  by 
Curlini,  111  ;  by  Airy,  113.  121  ; 
by  iMi'ndeidiall,  127  ;  by  Slerneck, 
128-131  ;  l.y  Luslia,  141  ;  Correc- 
tion for  Bunyjincy  of  Air  on.  First 
Used,  20  ;  (Jorrection  for.  Due  to 
Resisian(;e  nf  Air,  27,  00  ;  Methods 
of  Comparing  One  with  Another, 
113,  121,  128-130;  Balance,  131. 

Peters,  55.  124. 

Playfair,  55,  102. 

Plumb-line,  Deflection  of,  Observed 
at  Chimliorazo,  33-43  ;  at  Sche- 
hallien,  43,  53-50;  at  Arthur's 
Seal,  123;  at  Evaux.  118.  124; 
in  Tyrol.  124 ;  Deflection  of,  Cal- 
culated for  Chimborazo,  34;  how 
to  Observe.  35-39,  53 ;  by  Tides, 
44,  134,  135. 

Poisson,  31.  00   *• 

Power,  2-5.  40. 

Poynting,  10, 30, 44, 100. 112. 110, 118, 
119. 123-125,  127, 128.  131-134.  142. 

Pratt.  124. 

Pringle,  50. 

Puissant,  118. 

Pyramid,  Attraction  of  the  Great,  55. 

1 


Uelc'.!,  91.  100,  114-121,  125,  138. 
Kesislance  of  Air,  Discussed  by  Bon- 

guer.  27;    by   Cavendish,  05-07; 

by  I'oisson,  Alenubrea,  and  Cornu 

and  Bailie,  00,  100,  125. 
Itiehar/,  140,  141. 
Itobi.soii,  134. 
Koille,  48. 
Uoyal    Hoci  'y,     Experimeutn     by 

Members  ..f,  2-0,  48. 
Uo/.ier,  49. 


Sabine,  112. 

8aigey,  32,  43.  54,  112.  113.  118.  124; 

Corrt'ction  of  Peiiiviaii  Pendulum 

Experinients  by.  32,  43. 
Schaar,  119. 
Schettler.  123. 
Schehallien.  43,  53-55,  101, 112,  118, 

123 
SeheVl,  50.  112.  110,  118,  123. 
Sclnnidl,  44.  5.).  100,  112. 
Schubert.  124. 
Sheepshanks,  113. 
St.    Paul's   Cathedral,   Experiments 

at,  4,  5. 
Sterneck,  128-131,  137. 
Stokes,  122. 
Struve,  124,  134. 


Temperature,  Effects  of,  on  Torsion 
Balance,  Discussed  by  Cavendish, 
00,  70-80;  by  Reich.  114;  by 
Baily.  110,  117;  by  HicUs  and 
Poynting,  119  ;  by  Boys.  135.  130  ; 
by  E«)tv(is.  137  ;  by  Braiin,  139  ; 
Change  of  A  with.  125,  131  ; 
fjinni  of  Constancy  of,  135. 

Thiesen,  137. 

Thomson  arxl  Tait,  134. 

Tides,  Action  of,  on  Plumb-line,  44. 
134. 

Time  of  Vibration,  How  Found  by 
Cavendish.  04-07,  70  ;  by  Reich, 
115,  119  ;  by  Baily,  117  ;  by  Men- 
denhall.  127  ;  As  Affected  by  De- 
flection, 97  ;  As  AlTecied  i>y  Con- 
vection Currents,  80,  100,  134, 137  ; 
A  Found  From,  105,  100,  120, 
138,  139.  141. 

Todhunter,  10,  44.  50,  00. 
59 


u 


Ullott.  25,  39,  40. 


INJJEX 

Wilsing,  131,  182. 


Viiruum,  Experiment  Made  in,  138, 
141. 

W 

Wallenlin,  127.  134. 

We.HtininHUr    Ahhuy,    Experiments 

Miule  ut,  2.  8,  r». 
W  lie  well,  113. 


Young,  Kuleof,  31,  180. 

Z 

Zach.  3«.  44,  54-  56,  134  ;  Mnskelyne 

Exporiinenl    Culculaled    by,    64  ; 

Finds   Atlruction   of    Mount   Mi- 

Miet,  50. 
Zunotti  Bitmco,  44,  54.  50.  100,  112. 

113,  123,  127. 


ADDENDUM 

Page  32.  [Fut/c  (146J)  fum  calculated  the  diminution  in  the  attrnction  ac- 
cording to  hiii  fornuda  {see  note  on  p.  31),  and  Jinda  it  to  be  the  fi^i^fi  P'lrt, 
tehich  in  Hot  far  from  that  resulting  from  the  e^perime?it.  llin  calculation  can 
aim  be  stated  in  the  following  way:  taking  no  account  of  the  attraction  of  the 
plateau,  t/ie  observed  pendulum,  lengths  reduced  to  sea-level  by  Siiigey  are  at 

L'IsledeVInca 990.935  wj7m. 

Quito 991.009    " 

Difn'ence 074    " 

which  difference  is  of  the  orucr  of  the  errors  of  the  observations.  ^See  this  vol- 
ume, p.  130,  Ilelmert  (148.  vol.  2,  chap.  3),  and  Zanotti-Bianco  (li8^,pt.  1, 
ehap.  8,  a7idpt.  2,  p.  182).] 

160 


THE  END 


